Probab. Theory Relat. Fields 108, 459 ± 494 (1997)

Some central limit theorems for ``-valued semimartingales and their applications Yoichi Nishiyama The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan (e-mail: [email protected]) Received: 6 May 1996 / In revised form: 4 February 1997

Summary. This paper is devoted to the generalization of central limit the1 orems for empirical processes to several ÿ n;wtypes of
` …W†-valued continuousn time stochastic processes t , Xt ˆ Xt jw 2 W , where W is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t , Xtn;w is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic eciency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random ®elds of certain continuous semimartingales is derived. Mathematics Subject Classi®cation (1991): 60F05, 60F17, 62E20 1. Introduction This paper is devoted to the generalization of central limit theorems for empirical processes to several types of `1 …W†-valued continuous-time stochastic processes. Here W is a non-empty set, and we denote by `1 …W† the space of bounded real-valued functions de®ned on W. In order to explain the motivation, let us mention the results for empirical processes developed by Bolthausen [8], Dudley [10], [11], Gine and Zinn [13], [14], Ossiander [28] and Pollard [29], [30], among others. Let …E; E; P † be a probability space, and fZi g be an independent sequence of E-valued random variables identically distributed with P . For given family W of square-integrÿ able functions on E, consider the stochastic process X n ˆ Xtn;w j…t; w†
2 ‰0; 1Š  W de®ned by

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Y. Nishiyama

Xtn;w

 Z ‰ntŠ  1 X ˆ p w…Zi † ÿ w…z†P …dz† 8t 2 ‰0; 1Š 8 w 2 W : n iˆ1 E

…1†

Then, the central limit theorem says that, under appropriate conditions on the family W, the sequence of `1 …‰0; 1Š  W†-valued random elements X n converges in distribution to a Gaussian limit (but most authors considered only the terminal variables, i.e. t ˆ 1). Recently, those results are generalized to non-i.i.d. cases by several authors: to row-independent arrays by Alexander [2], Andersen et al. [4] and Pollard [31]; to stationary sequences by Arcone and Yu [5] and Doukhan et al. [9]; to stationary martingale di€erence arrays by Levental [25] and Bae and Levental [6]. See also the review by Wellner [34]. Now, ®x afresh any non-empty set W and a positive constant s. Set S ˆ ‰0; sŠ or fsg, and denote S ˆ S  W. In this paper, we establish some central limit theorems for `1 …W†-valued continuous-time stochastic processes t , Xtn ˆ …Xtn;w jw 2 W† as `1 …S†-valued random elements. To be more speci®c, we consider three kinds of situations as follows: (i) each coordinate process t , Xtn;w is a general semimartingale; (ii) each t , Xtn;w is represented as a stochastic integral, namely, ÿ Xtn;w ˆ W n;w  ln ÿ mn †t Z ˆ W n;w …x; s; x†…ln …x; ds; dx† ÿ mn …x; ds; dx†† ; ‰0;tŠE

where W n;w ˆ W n;w …x; t; x† is a predictable function on Xn  R‡  E, ln is an integer-valued random measure on R‡  E, and mn is the predictable compensator of ln ; (iii) each t , Xtn;w is a continuous local martingale. The results for (ii) and (iii) are proved by using the one for (i). The empirical processes are the special case of (ii). Sections 2.1, 2.2 and 2.3 are corresponding to (i), (ii) and (iii) above, respectively. The central limit theorem for 1-dimensional semimartingales was established by Liptser and Shiryaev [26], and has been expanded further into the more general limit theorems for ®nite-dimensional semimartingales, including the cases of non-Gaussian limits. We can consult the excellent books by Jacod and Shiryaev [22] and Liptser and Shiryaev [27] for those results, and use some of them to show the ®nite-dimensional convergence in our situations. On the other hand, many techniques to show the tightness are due to the previous works [4], [6], [13], [25] and [28] for empirical processes, and to Chap. 11 of Ledoux and Talagrand [24]. Our approach is essentially same as the one in empirical process theory, however, our framework and results are much broader. Further, new applications to statistical inference problems for stochastic processes are presented in Sects. 4 and 5. In Sect. 4, we consider the multiplicative intensity model for point processes with general marks. Let …E; E† be a Blackwell space on which a rÿ
®nite measure k is de®ned. For every n 2 N, let Tin ; Zin ; i 2 N be an Emarked point process, where 0 < T1n < T2n <


a.s. and each Zin is an E-

CLT for `1 -valued semimartingales

461

valued random variable. Suppose that its predictable compensator mn is of the form mn …x; dt; dz† ˆ a…t; z†Y n …x; t; z†dtk…dz† ; where a…t; z† is a non-negative real-valued measurable function on R‡  E, and Y n …x; t; z† is a non-negative real-valued predictable function on Xn  R‡  E. This model is a generalization of Aalen's multiplicative intensity model which is widely used in the context of survival analysis (see Aalen [1] and Andersen et al. [3]). We introduce certain class W of measurable functions on ‰0; sŠ  E, and consider the estimation problems for two kinds of `1 …W†-valued unknown parameters. One is the (weighted) cumulative hazard function A ˆ …A…w†jw 2 W† given by Z w…t; z†a…t; z† dtk…dz† ; …2† A…w† ˆ ‰0;sŠE

and the other is the (weighted) distribution function F ˆ …F …w†jw 2 W† given by  Z  Z F …w† ˆ w…t; z†a…t; z† exp ÿ a…s; x†dsk…dx† dtk…dz† : …3† ‰0;sŠE

‰0;tŠE

We introduce the generalized Nelson-Aalen's estimator for Eq. (2) and certain natural estimator for Eq. (3), and derive their asymptotic behavior by using the result in Sect. 2. We also prove their asymptotic eciency in the sense of convolution theorem. In Sect. 5, we consider certain regression model of continuous semimartingales. Let W be a family of locally bounded measurable functions on R‡ , and fan g be a sequence of positive constants such that an " 1. Let X n be a real-valued adapted 
process on a ®ltered probability space ÿ n n n continuous X ; F ; F ; Pn ˆ Phn ; h 2 W such that, under the probability measure Phn , it has the semimartingale decomposition given as follows: dXtn ˆ anh …t†Ztn Ytn dt ‡ dMtn;h

with

anh …t† ˆ a…t† ‡ aÿ1 n h…t† ;

where a ˆ a…t† is a locally bounded measurable function on R‡ , Y n and Z n are predictable and M n;h is a continuous local martingale with

n;h  R processes, t n n;h ˆ 0 Ys ds. We derive the M ;M t ÿ asymptotic
behavior of the log-likelihood ratio random ®elds t , Knt ˆ Kn …h†t jh 2 W where Kn …h†t ˆ log

dPhn jFnt : dP0n jFnt

See LeCam [23], Inagaki and Ogata [21] and Yoshida [35], among others, for such results in ®nite-dimensional parametric models. We apply it to construct a goodness of ®t test. Now, let us state some notations and de®nitions. When a pseudo-metric q is de®ned on S, we denote by Uq …S† the space of uniformly q-continuous real-valued functions on S. Convergence in distribution is considered in the Banach space `1 …S† with the sup-norm jj
jj1 . Since this space is not nec-

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Y. Nishiyama

essarily separable, we use the following de®nition of convergence in distribution due to Ho€mann-Jùrgensen and Dudley (see e.g. Ho€mann-Jùrgensen [19] or Sect. A.8 of Bickel et ÿal. [7]). Let
…X; dX † be a metric space. Let a n n n n? sequence ÿ
of probability spaces X ; F ; P be given. We denote by E n? resp: P the upper expectation (resp. the outer probability). De®nition 1.1. (i) For given sequence of mappings X n : Xn ! X (which are not necessarily Borel measurable), we say X n converges in distribution to the Borel probability measure L on …X; dX † if Z ÿ ÿ n

n? lim E f X f …x†L…dx† ˆ n!1

X

for every bounded dX -continuous function f : X ! R. We denote this converPn gence by ``X n ˆ) L in X''. (ii) For givenn sequence of mappings X n : Xn ! Y, where X  Y, the conP vergence ``X n ˆ) L in X'' ÿ is de®ned
to mean that P nthere exist mappings X~ n : Xn ! X such that P n? X n 6ˆ X~ n ˆ 0 and that X~ n ˆ) L in X in the sense of (i). Pn

P n?

We denote by X n ÿ! C (resp. X n ÿ! C) the convergence in P n -probability (resp. P n? -probability) when the limit C is deterministic. We follow [22] for notations and de®nitions in the theory of semimartingales. We need the following de®nitions to state some metric entropy conditions. (a) Let …S; q† be a totally bounded pseudo-metric space. De®nition 1.2. Given e > 0 we de®ne the e-covering number N …S; q; e† as the smallest number of closed balls, with centers in S and radius e in the pseudometric q, which form a covering of S. (b) Let W be a family of real-valued functions de®ned on a measurable space …E; E†. ÿ l;k u;k
De®nition 1.3. An E-measurable bracket for W is a family w ;w ; k 2 Kg of pairs of E-measurable functions such that: for every w 2 W there exists k 2 K such that wl;k  w  wu;k . We de®ne the size of the bracket by Card…K†. Further, in the situation (b), let . be a pseudo-metric de®ned on an appropriate subset of the space of E-measurable functions. De®nition 1.4. Given e > 0 we de®ne the e-bracketing ÿ l;k u;k
number N b …W; .; e† as ; w ; k 2 K for W such the smallest size of E-measurable brackets w
ÿ that . wl;k ; wu;k  e for every k 2 K.

2. Central limit theorems Recall the notation S ˆ S  W, where S ˆ ‰0; sŠ or fsg. All results in this section are available to both cases. Throughout this section, we shall assume

CLT for `1 -valued semimartingales

463

that there exists a pseudo-metric q on S which satis®es the following condition: …M† The space …S; q† is totally bounded and satis®es the metric entropy condition: Z 1 p log N …S; q; e† de < 1 : 0

Further, in the case of S ˆ ‰0; sŠ, it holds that: (i) q……t; /†; …s; /††  q……t; w†; …s; /†† for every …t; w†; …s; /† 2 S; (ii) the function t , q……t; /†; …s; /†† is continuous for every …s; /† 2 S; (iii) the function t , q……t; w†; …t; /†† is continuous and non-decreasing for every w; / 2 W. Remark 1. In the case of S ˆ fsg we may ignore the latter part of the condition …M†. In the case of S ˆ ‰0; sŠ, if the pseudo-metric spaces …S; .S † and …W; .W ) are totally bounded and satisfy the metric entropy condition respectively, and if the function t , .S …t; s† is continuous for every s 2 S, then either of the pseudo-metric q on S de®ned by q q ˆ .2S ‡ .2W or q ˆ maxf.S ; .W g satis®es the condition …M†.

2.1. Semimartingales ÿ
For every n 2 N, let Bn ˆ Xn ; Fn ; Fn ; P n be a ®ltered probability space. Let ÿ
X n ˆ X n;w jw 2 W be an RW -valued semimartingale de®ned on Bn ; that is, every coordinate process X n;w is a real-valued semimartingale on Bn . For every a > 0, let us introduce the truncation function ha …x† ˆ x1fjxjag . For every w 2 W, we consider the canonical decomposition of X n;w with respect to ha ; that is, X ÿ
 DXsn;w ÿ ha DXsn;w ; X n;w;a ˆ s


X

n;w

ÿ X n;w;a ˆ X0n;w ‡ Bn;w;a ‡ M n;w;a ;

…4†

where Bn;w;a is the predictable process with ®nite variation and M n;w;a is the locally square-integrable martingale such that n;w;a DB  a and DM n;w;a  2a (see 75±76pp. of [22]). The decomposition in Eq. (4) is unique up to a negstochastic processes ligible ÿ set, however, that the RW -valued
notice ÿ n;w;a
n;a n;w;a n;a jw 2 W and M ˆ M jw 2 W are not necessarily unique B ˆ B unless the set W is countable.

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Y. Nishiyama

Theorem 2.1. For every n 2 N, let X n be an RW -valued semimartingale such conditions …A1 )± for some
that X0n ˆ 0. Suppose the following ÿ
(A6 ) are satis®ed ÿ n  n;a ˆ X n;w ÿ X n;w;a jw 2 W into Bn;a ˆ Bn;w;a j w 2 W ÿ X decomposition of X ÿ
and M n;a ˆ M n;w;a jw 2 W : …A1 ) sup…t;w†2S Xtn;w …x† < 1 8x 2 Xn 8n 2 N; n;w;a P n?  …A2 ) supt2‰0;sŠ supw2W Xt 8a > 0; ÿ! 0 ÿ n;w;a
P n? …A3 ) supw2W Var B ÿ! 0 8a > 0; s  n;w n;/  P n …w;/† …A4 ) X ; X ÿ! Ct 8t 2 S 8w; / 2 W, t …w;/†

where Ct 's are real constants such that for every w; / 2 W the …w;/† is continuous on S; function t , Ct …A5 ) there exists a pseudo-metric q on S, satisfying the condition …M†, such that q …w;w† …/;/† …w;/† ‡ Cs ÿ 2Ct^s  q……t; w†; …s; /†† Ct 8…t; w†; …s; /† 2 S ;

…A6 ) for every e; g > 0 there exists a constant K > 0 such that: for every b > 0 there exists a constant a 2 …0; bŠ such that 0 1 lim sup P n? @ n!1

sup



t2S; w;/2W p q……t;w†;…t;/††K a

jMtn;w;a ÿ Mtn;/;a j > eA  g :

Then, it holds that:


ÿ (CLT) there exists a Borel probability measure G on `1 …S†; jj
jj1 , concentrated on the set Uq …S†, such that its ®nite-dimensional projection G…t1 ;w1 †;...;…td ;wd † is centered Gaussian distribution with  …w ;w † ÿ
Pn covariance matrix Cti ^ti j j , and that X n ˆ Xtn;w j…t; w† 2 S ˆ) G in `1 …S†. Remark 2. According to De®nition 1.1ÿ(ii), the condition (A1 ) can be slightly
weakened into the following form: P n? X n 62 `1 …S† ˆ 0 8n 2 N. Remark 3. In the case of S ˆ fsg, we may ignore the latter part of (A4 ) which …w;/† , as well as the latter part requires the continuity of each function t , Ct of …M†. This remark is available also to (B2 ) and (C2 ) below. The proof is given in Sect. 3.2. The condition …A6 † plays the same role as (3.2) in Theorem 3.1 of [13], and (ii) in Theorem 1 of [25]. It is not necessarily easy to check that condition, hence we will present more tractable conditions in two special cases below. 2.2. Weighted integer-valued random measures Let …E; E† be a Blackwell space. For every n 2 N, let ln be an integervalued random measure on R‡  E, de®ned on a ®ltered probability

CLT for `1 -valued semimartingales

465

ÿ
space Xn ; Fn ; Fn ; P n . Let mn be a ``good'' version of the predictable compensator of ln . See Sect. II.1 of [22] for the de®nitions. Here we assume that P n -a.s. …5† mn …‰0; sŠ  E† < 1 ÿ
n n;w Further, let a family W ˆ W jw 2 W of predictable functions on ~ n ˆ Xn  R‡  E be given. This means that every W n;w is a P ~ n -measurable X n n n n ~ ~ ˆ P E, and P denotes the predictable r-®eld function on X , where P on Xn  R‡ (see 16p. of [22]). We assume that there exists a predictable ~ n , which we call an envelope function of W n , such that function V n on X n;w ~n 8 w 2 W W …x; t; x†  V n …x; t; x† 8…x; t; x† 2 X and that V n  lns < 1 n 2

and

V n  mns < 1

P n -a.s.

…6†

n

and that the process jV j  m is locally integrable. It is clear that a sucient condition for Eq. (6) is En …V n  mns † < 1. Now we de®ne the RW -valued locally square-integrable martingale n X ˆ …X n;w jw 2 W† by X n;w ˆ W n;w  …ln ÿ mn †

8w 2 W :

…7†

Then the predictable quadratic covariation is given by X

n;w n;/  n;w n;/ n ˆ W W  m ÿ W^sn;w W^sn;/ ; X ;X t t W^tn;w …x†

st

R

where ˆ E W n;w …x; t; x†mn …x; ftg  dx†. ~ n such that Here, for given two predictable functions W n;…j† … j ˆ 1; 2† on X n;… j† 2 j  m is locally integrable, we de®ne each jW q 2 n;…2† qn …W n;…1† ; Wt † ˆ jW n;…1† ÿ W n;…2† j mnt : Then it is trivial that q hX n;…1† ÿ X n;…2† ; X n;…1† ÿ X n;…2† it  qn …W n;…1† ; W n;…2† †t ; where X n;… j† ˆ W n;… j†  …ln ÿ mn †. Theorem 2.2. In the situation stated above, consider the RW -valued local martingale X n ˆ …X n;w jw 2 W† de®ned by Eq. …7†. Then, …CLT† holds under the following conditions …B1 †±…B5 †: Pn

…B1 † jV n j2 1fV n >eg  mns ÿ! 0 n;w

n;/

Pn

…w;/† Ct

8 e > 0;

…B2 † hX ; X it ÿ! 8 t 2 S 8 w; / 2 W, …w;/† where Ct 's are real constants such that for every w; / 2 W the …w;/† is continuous on S; function t , Ct …B3 † the condition …A5 † in Theorem 2.1 is satis®ed; …B4 † there exists a constant L1 > 0 such that

466

Y. Nishiyama

0 lim lim sup P n? @ sup b#0

n!1

t2S; w;/2W q……t;w†;…t;/††b

1 ÿ
qn W n;w ; W n;/ t > L1 bA ˆ 0 ;

…B5 † there exists a constant L2 > 0 such that: for every n 2 N and e > 0 ~ n -measurable bracket for W n ˆ …W n;w jw 2 W† with there exists a P ®nite size Nb …n; e†, namely f…W n;e;l;k ; W n;e;u;k †; 1  k  Nb …n; e†g, such that ! ÿ
qn W n;e;l;k ; W n;e;u;k s n? > L2 ˆ 0 sup max lim lim sup P b#0 n!1 e 0
Z lim lim sup b#0

n!1

0

b

p log Nb …n; e† de ˆ 0 :

The proof is given in Sect. 3.3. The condition …B5 † is a generalization of the metric entropy condition with L2 -bracketing due to [28] (see also Proposition 1.1 of [6]). That condition may seem to be arti®cial at ®rst sight, but it is weak enough for applications. We can derive, for instance, the asymptotic behavior of the Nelson-Aalen's estimator for marked point processes under quite natural conditions (Theorem 4.3 below). That result is interesting by itself in the context of survival analysis. Now, let us brie¯y see that Ossiander's central limit theorem [28] for i.i.d. sequences is also deduced from Theorem 2.2, in order to illustrate how to apply our theorem to discrete-time stochastic processes. Let …E; E; P † be a probability space, and set n X ei=n …dt†P …dx† ; mn …x; dt; dx† ˆ iˆ1

where ea denotes the Dirac measure at point a. For given family W of squareintegrable functions on E, we set w…x† W n;w …x; t; x† ˆ p n

8w2W :

Then the stochastic process X n de®ned by Eq. (7) coincides with the empirical process Eq. (1). Here assume that the family W has a square-integrable envelope function u and satis®es s Z Z 1 p jw…x† ÿ /…x†j2 P …dx† : log Nb …W; .; e† < 1 with .…w; /† ˆ 0

E

Then all of the conditions in Theorem 2.2 are ful®lled if we de®ne the pseudo-metric q on S ˆ ‰0; 1Š  W by s Z q……t; w†; …s; /†† ˆ jt ÿ sj ju…z†j2 P …dz† ‡ j.…w; /†j2 : E

CLT for `1 -valued semimartingales

467

It is clear from the observation above that we can obtain the results for empirical processes with random weights in quite general situations, including dependent cases. But we do not pursue them here in concrete fashions.

2.3. Continuous local martingales For every n 2 N, let X n ˆ …X n;w jw 2 W† be an RW -valued continuous local martingale de®ned on a ®ltered probability space …Xn ; Fn ; Fn ; P n †. For given w; / 2 W we de®ne q cn …w; /†t ˆ hX n;w ÿ X n;/ ; X n;w ÿ X n;/ it : Theorem 2.3. For every n 2 N, let X n be an RW -valued continuous local martingale such that X0n ˆ 0. Then, (CLT) holds under the following conditions …C1 †±…C5 †: …C 1 † the condition …A1 † in Theorem 2.1 is satis®ed;

Pn …w;/† 8 t 2 S 8 w; / 2 W, …C 2 † X n;w ; X n;/ it ÿ! Ct …w;/† where Ct 's are real constants such that for every w; / 2 W the …w;/† function t , Ct is continuous on S; …C 3 † the condition …A5 † in Theorem 2:1 is satis®ed for a proper metric q de®ned on S; …C 4 † each process X n is separable with respect to the metric q in the following sense: there exist a countable subset Wn  W and a negligible set N n  Xn such that for every x 62 N n , …t; w† 2 S and e > 0  Xtn;w …x† 2 Xtn;/ …x†; / 2 Wn ; q……t; w†; …t; /†† < e ; where the closure is taken in R; …C 5 † there exists a constant L > 0 such that 0 1 n c …w; /†t > LA ˆ 0 : lim lim sup P n @ sup b#0 n!1 q……t; w†; …t; /†† t2S; w;/2Wn 0
Remark 4. The condition …C3 † can be slightly weakened into the following form: the condition …A5 † in Theorem 2.1 is satis®ed for a pseudo-metric q de®ned on S such that q……t; w†; …t; /†† ˆ 0 implies w ˆ / [i.e., q……t;
†; …t;
†† is a proper metric on W for every t 2 S]. The proof is given in Sect. 3.4. The condition …C5 † is similar to …B4 † and …B5 †. But we require a proper metric q in …C3 † and the separability of stochastic processes in …C4 † instead of the brackets, because the latter one does not work well for the processes which do not have ®nite variation. An application of the result above is presented in Sect. 5.

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3. Proofs 3.1. Auxiliaries The following fact is the starting point of our results (see e.g. Theorem 10.2 of Pollard [31] for the proof ). Lemma 3.1. Let …S; q† be a totally bounded pseudo-metric space. For every n 2 N, let X n ˆ …X n …s†js 2 S† be an `1 …S†-valued random element de®ned on a probability space …Xn ; Fn ; P n †. Suppose the following conditions …i† and …ii†: …i †

the ®nite-dimensional distributions converge, that is, for every d 2 N and every s1 ; . . . ; sd 2 S there exists a Borel probability measure Ls1 ;...;sd on …Rd ; Rd † such that Pn

…X n …s1 †; . . . ; X n …sd †† ˆ) Ls1 ;...;sd

in

Rd ;

…ii† for every e; g > 0 there exists a constant d > 0 such that ! lim sup P n? n!1

sup jX n …s† ÿ X n …t†j > e

q…s;t†
g :

Then, there exists a Borel probability measure L on …`1 …S†; jj
jj1 †, concenits ®nite-dimensional projections are Ls1 ;...;sd trated on the set Uq …S†, such that Pn n appeared in (i), and that X ˆ) L in `1 …S†. Here are some useful inequalities (see e.g. Sect. 4.13 of [27] for the proof ). Lemma 3.2. (i) If X is a real-valued local martingale such that X0 ˆ 0 and that jDX j  a for some a > 0, then it holds that for every t; C; e; k > 0 !   exp…ka† ÿ 1 ÿ ka P sup jXs j > e; h X ; X it  C  2 exp ÿke ‡ C : a2 s2‰0;tŠ …ii† If X is a real-valued continuous local martingale such that X0 ˆ 0, then it holds that for every t; C; e > 0 !   e2 : P sup jXs j > e; hX ; X it  C  2 exp ÿ 2C s2‰0;tŠ We shall use the preceding lemma in the following forms, in which (a) is the Bernstein's inequality for continuous-time local martingales with bounded jumps. Corollary 3.3. Let X ; Y be real-valued local martingales such that X0 ˆ Y0 ˆ 0 and that jDX j  a; jDY j  a for some a > 0. …a†

It holds that for every t; e; C > 0

CLT for `1 -valued semimartingales

469

! P …b†

sup jXs j > e; hX ; X it  C

s2‰0;tŠ

e2 2‰ae ‡ CŠ

 :

If 0  e  Ca then it holds that for every t > 0 ! P

…c†

  2 exp ÿ

sup jXs j > e; hX ; X it  C

s2‰0;tŠ

  e2 :  2 exp ÿ 4C

C If 0  e  2a then it holds that for every t; s > 0

ÿ

P jXt ÿ Ys j > e; hX ; X it ‡ hY ; Y is ÿ 2hX ; Y it^s

 2 e  C  2 exp ÿ : 4C


Proof. By the same calculation as Freedman [12], the inequality (a) is easily derived from (i) of Lemma 3.1. Next, the assertion (b) is immediate ÿfrom
(a). In order to show (c) it is enough to apply (b) to the process Z ˆ Zu u2R‡ , where Zu ˆ Xu^t ÿ Yu^s . (

3.2. Proof of Theorem 2.1 Let us begin with two lemmas, in which we use the notation cn;a ……t; w†; …s; /†† q ˆ hM n;w;a ; M n;w;a it ‡hM n;/;a ; M n;/;a is ÿ2hM n;w;a ; M n;/;a it^s : Lemma 3.4. Assume the conditions …A2 †±…A5 †, and ®x any ®nite subset Sf  S. Then, it holds that for every a > 0 and d > max…t;w†;…s;/†2Sf q……t; w†; …s; /††   n n;a max c ……t; w†; …s; /†† > d ˆ 0 : lim P n!1

…t;w†;…s;/†2Sf

Proof. By Theorem VIII.3.6 and Lemma VIII.3.16 of [22], it holds that for every a > 0

n;w;a n;/;a  P n …w;/† ;M ÿ! Ct 8 t 2 S 8 w; / 2 W : M t Thus for every …t; w†; …s; /† 2 Sf q Pn …w;w† …/;/† …w;/† cn;a ……t; w†; …s; /†† ÿ! Ct ‡ Cs ÿ 2Ct^s  q……t; w†; …s; /†† : This fact yields the claim.

(

The next lemma plays the key role in our context. In the proof we shall follow the chaining argument which is used in those for Theorem 3.2 of [13] and Theorem 2 of [25]. See also Sect. VII.6 of [30].

470

Y. Nishiyama

Lemma 3.5. Assume the conditions …A2 †±…A5 †. Then, for every e; g; K > 0 there exist some constants d; b > 0 such that for every  > 0 and a 2 …0; bŠ ! lim sup P n? n!1

sup

q……t;w†;…s;/††
jMtn;w;a ÿ Msn;/;a j > e ‡ 2

0

 g ‡ lim sup P n? @ n!1

1 sup



t2S; w;/2W p q……t;w†;…t;/††K a

jMtn;w;a ÿ Mtn;/;a j > A :

Proof. Let e; g; K > 0 be given, and ®x p any J > 1. Set R ˆ K=2 and   D ˆ 10 _ …e=4J 2 R2 †. We denote H …x† ˆ log…N …x†=x†, where N …x† ˆ N …S; q; x†. Here notice that Z xH …x† 

x 0

Z H …y† dy

and

lim x#0

0

x

H …y† dy ˆ 0 :

Hence we can ®nd a constant x0 > 0 such that the following inequalities hold for any x 2 …0; x0 Š: Z x e …8† 8J H …y† dy  ; 5 0 xH …x† 

JR2 ; 2

   x2 8 e2 g exp 2 jx=2j2 H …x=2†2 ÿ  ; 2 20000 x 3     1 e2 g  : 4x exp 2 x2 H …x†2 ÿ 2 2 16D J x 3

…9† …10† …11†

p Then, we set ÿ p
d ˆ …g=6† ^ x0 and de®ne the constant b > 0 by R b ˆ R 125d2 =2e ^ d. For p every a 2 …0; bŠ, let us choose ka 2 N [ f0g p such that d=2 < 2ka R a  d. For every i ˆ 0; 1; . . . ; ka , setting dai ˆ 2i R a, let us ®x a subset Sai  S consists of the centers of closed balls which form a dai -covering of S, such that Card…Sai † ˆ N …dai †. Then, we introduce the mappings kai : S ! Sai such that q……t; w†; kai …t; w††  dai

8…t; w† 2 S :

Further we de®ne the mappings tia ˆ pS  kai  kaiÿ1 


 ka0

and

wai ˆ pW  kai  kaiÿ1 


 ka0 ;

where pS (resp. pW ) is the projection from S to S (resp. W ). Now, observe that for every …t; w†; …s; /† 2 S n;w;a M t ÿ Msn;/;a  …a† ‡ …b† ‡ …c† ‡ …d† ; where each term in the right hand side is given by:

CLT for `1 -valued semimartingales

471

a n;wa …t;w†;a (a) ˆ Mtn;w;a ÿ Mt 0 ‡ Msn;/;a ÿ Msn;w0 …s;/†;a ; n;wa …t;w†;a a n;wa0 …t;w†;a n;wa0 …s;/†;a ÿ Mta …t;w† (b) ˆ Mt 0 ‡ Msn;w0 …s;/†;a ÿ Mta …s;/† ; 0

0

ka X n;waiÿ1 …t;w†;a n;wai …t;w†;a ÿ Mta …t;w† (c) ˆ Mta …t;w†

‡

i

iÿ1

iˆ1

ka X n;waiÿ1 …s;/†;a n;wai …s;/†;a ÿ Mta …s;/† Mta …s;/† ; iˆ1

i

iÿ1

n;waka …s;/†;a n;waka …t;w†;a ÿ Mta …s;/† (d) ˆ Mta …t;w† : ka

ka

So it holds that for every  > 0 P

n?

sup

q……t;w†;…s;/††
! n;w n;/ Xt ÿ Xs > e ‡ 2  (I) ‡ (II) ‡ (III) ‡ (IV) ;

where each term in the right hand side is given by: ! n;wa0 …t;w†;a n;w;a sup Mt ÿ Mt > ;

(I) ˆ P

n?

(II) ˆ P

n

e n;wa0 …t;w†;a n;wa …t;w†;a sup Mt 0 ÿ Mta …t;w† > 0 5 …t;w†2S

(III) ˆ P

n

ka e X n;waiÿ1 …t;w†;a n;wai …t;w†;a sup ÿ Mta …t;w† Mta …t;w† > i iÿ1 5 …t;w†2S iˆ1

…t;w†2S

(IV) ˆ P n

! ; !

e n;waka …s;/†;a n;waka …t;w†;a sup ÿ Mta …s;/† Mta …t;w† > ka ka 5 q……t;w†;…s;/††
; ! :

Here notice that the term (II) vanishes in the case of S ˆ fsg. The following arguments about the terms (I), (III) and (IV) are available to both of the cases S ˆ ‰0; sŠ and fsg. Estimation of (I). It follows from (i) in the condition …M† that q……t; w†; …t; wa0 …t; w†††  q……t; w†; ka0 …t; w†† ‡ q…ka0 …t; w†; …t; wa0 …t; w†††  q……t; w†; ka0 …t; w†† ‡ q…ka0 …t; w†; …t; w†† p p  2da0 ˆ 2R a ˆ K a :

Hence we obtain

0

1 (I)  P n? @ sup Mtn;w;a ÿ Mtn;/;a > A : t2S; w;/2W p q……t;w†;…t;/††K a

472

Y. Nishiyama

ÿ
Estimation of (III). For every i ˆ 1; 2; . . . ; ka we set eai ˆ 2 J dai H daiÿ1 . Then it holds that ka X iˆ1

eai

ˆ 8J

ka X da

iÿ1

iˆ1

2

H

ÿ

daiÿ1




Z  8J

0

d

H …x† dx 

e ; 5

where the last inequality follows from Eq. (8). Next it is immediate from Eq. (9) that 2 p ÿ p ÿ p aH R a† J dai jJ dai j2 a a
 : e i  2 J di H R a † ˆ 2a 2a 2iÿ2 JR n Here we de®ne the set Xn;a 1 2 Fs by 8 9 > > k < = a \ n;a a n;a X1 ˆ : max c ……t; w†; …s; /††  J d i >…t;w†2Saiÿ1 ; …s;/†2Sai > ; iˆ1: q……t;w†;…s;/††
Then we obtain by using Corollary 3.3 (c) that P

n

ka e X n;waiÿ1 …t;w†;a n;wai …t;w†;a n;a sup ÿ Mta …t;w† Mta …t;w† > ; X1 i iÿ1 5 …t;w†2S iˆ1



ka X iˆ1

 ˆ

P

n

n;waiÿ1 …t;w†;a n;wai …t;w†;a n;a sup Mta …t;w† ÿ Mta …t;w† > eai ; X1

…t;w†2S

iÿ1

ka X ÿ
jea j2 N daiÿ1
2 exp ÿ i a 2 4jJ di j iˆ1 ka X iˆ1

!

2daiÿ1  2d 

!

!

i

g ; 3

where the last ÿ inequality
follows from the de®nition of d. So we have (III)  g3 ‡ P n Xn n Xn;a 1 . Estimation of (IV). Note that if q……t; w†; …s; /†† < d then q……tkaa …t; w†; waka …t; w††; …tkaa …s; /†; waka …s; /†††  q……t; w†; …s; /††

‡ q……t; w†; …t0a …t; w†; wa0 …t; w††† ‡ q……s; /†; …t0a …s; /†; wa0 …s; /†††

‡

ka X iˆ1

‡

ka X iˆ1

a q……tiÿ1 …t; w†; waiÿ1 …t; w††; …tia …t; w†; wai …t; w††† a q……tiÿ1 …s; /†; waiÿ1 …s; /††; …tia …s; /†; wai …s; /†††

< d ‡ 4daka ÿ 2da0 < 5d :

CLT for `1 -valued semimartingales

473

n So we de®ne the set Xn;a 28 2 Fs by 9 < = n;a Xn;a ˆ max c ……t; w†; …s; /††  5d : 2 : …t;w†;…s;/†2Saka ; q……t;w†;…s;/††<5d

2

It is immediate from the de®nition of b that 5e  j5dj 2a . Hence we obtain by using Corollary 3.3 (c) that ! e a a n;w …t;w†;a n;w …s;/†;a n;a ka ka Pn sup ÿ Mta …s;/† Mta …t;w† > ; X2 ka ka 5 q……t;w†;…s;/††
ta0 …t; w† ˆ supfu 2 S; q……u; wa0 …t; w††; …t0a …t; w†; wa0 …t; w†††  da0 g :

Then it follows from (ii) in the condition …M† that

p q……ta0 …t; w†; wa0 …t; w††; …ta0 …t; w†; wa0 …t; w†††  2da0 < 2 JR a :

n So we de®ne the set Xn;a 3 2 Fs by (

Xn;a 3

ˆ

sup c

…t;w†2S

n;a

……ta0 …t; w†; wa0 …t; w††; …ta0 …t; w†; wa0 …t; w†††

p  2 JR a

) :

p 2 It is immediate from the de®nition of D that De  j2 JRa aj . Hence we obtain by using Corollary 3.3(b) that for every …t; w† 2 S ! e n;wa0 …t;w†;a n;wa0 …t;w†;a n;a n sup ÿ Mta …t;w† > ; X3 P Ms a 0 10 s2‰ ta0 …t;w†;t0 …t;w†Š ! e n;wa0 …t;w†;a n;wa0 …t;w†;a n;a n sup ÿ Mta …t;w† > ; X3  P Ms a 0 D s2‰ ta0 …t;w†;t0 …t;w†Š ! je=Dj2 :  2 exp ÿ p 4j2 JR aj2

It is clear from (i) in the condition …M† that ta0 …t; w†  t  ta0 …t; w†, hence it follows from the above inequality that

474

Y. Nishiyama

P

e n;wa …t;w†;a n;wa0 …t;w†;a n;a sup Mt 0 ÿ Mta …t;w† > ; X3 0 5 …t;w†2S

n

 P

n

!

e n;wa0 …t;w†;a n;wa …t;w†;a n;a sup Mt 0 ÿ Mta …t;w† > ; X3 0 10 …t;w†2S

!

! e n;wa0 …t;w†;a n;wa0 …t;w†;a n;a sup Mta …t;w† ÿ Mta …t;w† > ; X3 ‡P 0 0 10 …t;w†2S   p e2  2N …R a†
2 exp ÿ 2 16D J 2 R2 a   p p e2 ˆ 4R a exp H …R a†2 ÿ 16D2 J 2 R2 a    p p 2 1 e2 g 2  ˆ 4R a exp 2 R a
H …R a† ÿ ; 16D2 J 2 R a 3 n

ÿ
where the last inequality follows from Eq. (11). So (II)  g3 ‡ P n Xn n Xn;a 3 . By Lemma 3.4 it holds that limn!1 P n …Xn n Xn;a j † ˆ 0 for j ˆ 1; 2; 3. So the assertion of the lemma follows from the estimates of (I), (II), (III) and (IV). ( Proof of Theorem 2.1. We shall check the conditions in Lemma 3.1. The convergence of ®nite-dimensional distributions follows from the usual semimartingale central limit theorem (see Theorem VIII.3.6 and Lemma VIII.3.16 of [22]) using …A2 †±…A4 †. To show (ii) in Lemma 3.1, ®rst notice that for every …t; w†; …s; /† 2 S ( ) ‡ Mtn;w;a ÿ M n;/;a : jXtn;w ÿ X n;/ j  2 sup Xtn;w;a ‡ sup Bn;w;a t s

…t;w†2S

s

…t;w†2S

For given e; g > 0 choose K > 0 according to …A6 †. Next, choose d; b > 0 according to Lemma 3.5. Finally, if we choose a 2 …0; bŠ such that 0 1 n;w;a n;/;a M t ÿ Mt > eA  g ; lim sup P n? @ sup t2S; w;/2W n!1 p q……t;w†;…t;/††K a then it follows from …A2 † and …A3 † that lim sup P

n?

n!1

sup

q……t;w†;…s;/††
 lim sup P n? n!1

‡ lim sup P n!1

n;w Xt ÿ X n;/ > 7e

!

s

>e sup Xtn;w;a > e; sup Bn;w;a t

…t;w†2S n?

…t;w†2S

sup

q……t;w†;…s;/††
n;w;a M t ÿ M n;/;a > 3e s

!

!  2g :

CLT for `1 -valued semimartingales

475

This completes the proof.

(

3.3. Proof of Theorem 2.2 Throughout this subsection, we will use the following notations: for every n; p; r 2 N and a  1 v u r Y u n Nb …n; 2ÿj † ; fr ˆtlog 2r jˆ1

enp ˆ

1 X rˆp

ga ˆ sup

2ÿr fnr

;

1 r X Y

n2N rˆ1

ÿ

Nb n; 2

ÿj

!


jˆ1

ÿ
exp ÿ ajfnr j2 :

Here Nb …
;
† denotes the integer which appeared in the condition …B5 †. To prove Theorem 2.2 we shall check the conditions in Theorem 2.1. The following three lemmas concern the condition …A6 †. The idea to show the ®rst one is taken from [4]. Lemma 3.6. Under the condition …B5 † it holds that: (i ) lima!1 ga ˆ 0 ; (ii ) limp!1 lim supn!1 enp ˆ 0

:

Proof. Observe that 1 r Y X rˆ1

ÿ

Nb n; 2

ÿj




!

jˆ1

ˆ

1 X

2

ÿar

rˆ1

 ˆ

1 X

r Y

ÿ
exp ÿ ajfnr j2 ÿ

Nb n; 2

ÿj




!1ÿa

jˆ1

2ÿar whenever a  1

rˆ1 ÿa

2 ; 1 ÿ 2ÿa

which converges to 0 as a ! 1. So we obtain (i). To prove (ii) we set nnj ˆ Since fnr 

Pr

jˆ1

p log 2Nb …n; 2ÿj † :

nnj we have for every integer p  2

476

Y. Nishiyama 1 X rˆp

2ÿr fnr  ˆ

1 X

2ÿr

r X

rˆp

jˆ1

1 X

1 X

jˆ1

nnj

 2ÿp‡1

nnj 2ÿr

rˆj_p

‰p=2Š X jˆ1

nnj ‡

 2ÿp‡1‡‰p=2Š

‰p=2Š X jˆ1

1 X jˆ‰p=2Š‡1

2ÿj‡1 nnj

2ÿj nnj ‡ 2

1 X jˆ‰p=2Š‡1

2ÿj nnj :

The condition …B5 † implies lim sup n!1

1 X jˆ1

2ÿj nnj < 1 and lim lim sup p!1

n!1

1 X jˆ‰p=2Š‡1

2ÿj nnj ˆ 0 ;

hence we obtain (ii).

(

By virtue of (ii) in the above lemma, we have in particular that: there exist p0 ; n0 2 N such that p  p0 ; n  n0 ˆ) 2ÿp fnp 

1 : 2

…12†

Based on this fact, let us state Lemma 3.7 which plays the key role in our context. The bracketing argument in the proof is basically due to [28]. It is shown by [6] that certain martingale inequality, which is Corollary 3.3 (a) above, works well instead of the Bernstein's inequality in the i.i.d. case. See van de Geer [32] for a generalization of Corollary 3.3 (a) and its application. Lemma 3.7. Fix any a  1 and n; p; q 2 N such that p0  p < q and n  n0 , where p0 ; n0 are the integers appeared in Eq. (12 ). De®ne the local martingale M n;w;…p† , the set Xna;p;q 2 Fns and the stopping time snq in the following way: M n;w;…p† ˆ W n;w 1fV n 2ÿ2p g  …ln ÿ mn † ; ) ÿ
ÿj ÿj qn W n;2 ;l;k ; W n;2 ;u;k s p n Xa;p;q ˆ max max  a ; pjq 1kNb …n;2ÿj † 2ÿj n o snq ˆ inf t 2 R‡ ; mn …‰0; tŠ  E†  4jfnq‡1 j2 ÿ 1 ^ s : (

Then, it holds that for every  > 0

…13†

…14†

CLT for `1 -valued semimartingales

0 P n? @

sup

t2S; w;/2W q……t;w†;…t;/††2ÿp

477

1 n;w;…p† n;/;…p† ÿ Mt Mt > 20aenp ‡ A

 P n …Xn n Xna;p;q † ‡ P n …snq < s† ‡ 2ga=4 ‡ 2ga=10    2 2p ÿ2p‡1 ÿp ‡ 2 exp 2 2 log Nb …n; 2 † ÿ 4 ‡ 18a 0 1 p   ‡ P n? @ sup qn …W n;w ; W n;/ †t > 2ÿp aA : t2S; w;/2W q……t;w†;…t;/††2ÿp

Before the proof of above lemma, let us show that: Lemma 3.8. Under the conditions …B4 † and …B5 †, it holds for every e > 0 0 1 n;w;…p† n;/;…p† lim lim sup P n? @ sup Mt ÿ Mt > eA ˆ 0 ; p!1 n!1

t2S; w;/2W q……t;w†;…t;/††2ÿp

where M n;w;…p† is de®ned by Eq. (13 ). Proof. Fix any ; g > 0. By Lemma 3.6 p(i)  we can choose a large constant a such that 2ga=4 ‡ 2ga=10  g and that a  maxf1; L1 ; L2 g, where L1 and L2 are the constants in the conditions …B4 † and …B5 †. On the other hand, since p fnq  log 2q , it holds for the stopping times snq de®ned by Eq. (14) that for every n 2 N   P n …snq < s†  P n mn …‰0; sŠ  E†  4jfnq‡1 j2 ÿ 1  P n …mn …‰0; sŠ  E†  4…q ‡ 1† log 2 ÿ 1† ; and the right hand side converges to zero as q ! 1 because of Eq. (5). Further, observe that p lim lim sup 2ÿp log Nb …n; 2ÿp † ˆ 0 ; p!1

n!1

and thus

  lim lim sup exp 22p 2ÿ2p‡1 log Nb …n; 2ÿp † ÿ

p!1

n!1

2 4 ‡ 18a

 ˆ0 :

Hence, it follows from the estimate in Lemma 3.7 that 0 1 n;w;…p† n;/;…p† ÿ Mt lim sup lim sup P n? @ sup Mt > 20aenp ‡ A p!1

n!1

t2S; w;/2W q……t;w†;…t;/††2ÿp

 lim sup lim sup lim sup P n …Xn n Xna;p;q † ‡ g ˆ g : p!1

n!1

q!1

478

Y. Nishiyama

This fact and Lemma 3.6 (ii) yield the assertion.

(

Proof of Lemma 3.7. For every n; j 2 N, let us introduce a mapping k…
; n; j† : W ! f1; 2; . . . ; Nb …n; 2ÿj †g such that W n;2

ÿj

;l;k…w;n;j†

ÿj

 W n;w  W n;2

;u;k…w;n;j†

:

For every w 2 W and every integer r such that p  r  q, we de®ne the n;w;r ~ n by on X predictable functions W n;w;r and W " # r _ n;w;r n n;2ÿj ;l;k…w;n;j† W ˆ …ÿV † _ W 1fV n 2ÿ2p g jˆp

and

" W

n;w;r

n

ˆ V ^

r ^ jˆp

# W

n;2ÿj ;u;k…w;n;j†

1fV n 2ÿ2p g :

Here notice that if r1  r2 then W n;w;r1  W n;w;r2  W n;w 1fV n 2ÿ2p g  W

n;w;r2

W

n;w;r1

:

…15†

~ n given by Next we introduce the predictable subsets of X n o n;w;r ÿ W n;w;r > …2r‡1 fnr‡1 †ÿ1 : Anr …w† ˆ W Now, for every w 2 W we consider the disjoint partition fBnr …w†; p  r  qg ~ n de®ned by of predictable subsets of X 8 n for r ˆ p , > < Ap …w†; S rÿ1 n n n Br …w† ˆ Ar …w† n jˆp Aj …w†; for p ‡ 1  r  q ÿ 1 , > :X ~ n n Sqÿ1 Bn …w†; for r ˆ q . jˆp j We deal with three types of local martingales given as follows: M n;w;…p† ˆ W n;w 1fV n 2ÿ2p g  …ln ÿ mn † q X W n;w 1fV n 2ÿ2p g 1Bnr …w†  …ln ÿ mn † ; ˆ rˆp

Ln;w;p;q ˆ

q X rˆp

W n;w;r 1Bnr …w†  …ln ÿ mn † ;

Ln;w;…p† ˆ W n;w;p  …ln ÿ mn † q X W n;w;p 1Bnr …w†  …ln ÿ mn † : ˆ rˆp

Then it holds that

CLT for `1 -valued semimartingales

0 P n? @ P

sup

t2S;w;/2W q……t;w†;…t;/††2ÿp

479

1 n;w;…p† n;/;…p† ÿ Mt M t > 20aenp ‡ ; Xna;p;q A

n;w;…p† sup Mt ÿ Ltn;w;p;q > 9aenp ; Xna;p;q

n?

…t;w†2S

‡P

n;w;…p† sup Ltn;w;p;q ÿ Lt > aenp ; Xna;p;q

n?

…t;w†2S

0 ‡ P n? @

sup

t2S; w;/2W q……t;w†;…t;/††2ÿp

! !

1 n;w;…p† n;/;…p† ÿ Lt Lt > ; Xna;p;q A

ˆ …I† ‡ …II† ‡ …III† : Hereafter we shall estimate (I), (II) and (III). Estimation of (I). Observe that n;w;…p† M ÿ Ln;w;p;q 

q X ÿ rˆp

‡ 


W n;w 1fV n 2ÿ2p g ÿ W n;w;p;r 1Bnr …w†  ln

q X ÿ

rˆp q X


W n;w 1fV n 2ÿ2p g ÿ W n;w;r 1Bnr …w†  mn

W

n;w;r

rˆp

‡

q  X rˆp

ˆ

q X

W

 ÿ W n;w;r 1Bnr …w†  ln

n;w;r

 ÿ W n;w;r 1Bnr …w†  mn

U n;w;r  …ln ÿ mn † ‡ 2

q X

rˆp

U n;w;r  mn ;

…16†

rˆp

  n;w;r ÿ W n;w;r 1Bnr …w† . Let us consider the last term in the where U n;w;r ˆ W right hand side of Eq. (16) at ®rst. For p  r  q ÿ 1, it is clear from Bnr …w†  Anr …w† that jU n;w;r j2  …2r‡1 fnr‡1 †ÿ1
U n;w;r , hence we have ( ) qÿ1 qÿ1 X X n;w;r n ÿ…r‡1† n U  ms > 4a 2 fr‡1 sup w2W rˆp

(

 ( 

rˆp

n;w;r

U  mn max sup ÿr‡1 n s > a prqÿ1 w2W 2 fr‡1

)

jU n;w;r j2  mns max sup >a prqÿ1 w2W 2ÿ2r

)  Xn n Xna;p;q

:

480

Y. Nishiyama

As for r ˆ q, since mn …‰0; snq Š  E†  4jfnq‡1 j2 , it follows from the Schwarz's inequality that ( ) sup U n;w;q  mnsnq > 4a
2ÿ…q‡1† fnq‡1

w2W

(

) sup jU

 ( 

n;w;q 2

j 

w2W

sup

\

snq

…‰0; snq Š

>a

(

o

n

ˆ s  sup

2

q X

w2W rˆp

( 

 E† > j4a


2ÿ…q‡1† fnq‡1 j2

)

2ÿ2q

w2W

n


m

jU n;w;q j2  mnsnq

From these facts we get Xna;p;q

mnsnq

sup 2

 Xn n Xna;p;q

U

n;w;r

q X

…t;w†2S

U



mns

n;w;r

 4a

…recall a  1† :

q X rˆp



rˆp

mnt



) 2ÿ…r‡1† fnr‡1 )

8aenp

;

which implies that q X n;w;r U sup  …ln ÿ mn †t > aenp ; Xna;p;q

(I)  P n?

! …17†

…t;w†2S rˆp

  ‡ P n snq < s : Here, for every p  r  q, it holds that jU n;w;r j2  mns  2ÿ2r a and that

on the set

Xna;p;q

ÿ
ÿ1 0  U n;w;r  2r fnr

n ÿ[asn for
r cˆ p, use Eq. (12); as for p ‡ 1  r  q, notice that Br …w†  Arÿ1 …w† and use Eq. (15)]. So Corollary 3.3 (a) yields that ! P n sup U n;w;r  …ln ÿ mn † > 2ÿr fn a; Xn t

…t;w†2S

 2 exp ÿ

r

a;p;q

!

j2ÿr fnr aj2

2‰…2r fnr †ÿ1
2ÿr fnr a ‡ 2ÿ2r aŠ

ajfn j2 ˆ 2 exp ÿ r 4

! :

Now, it is clear that r  Y ÿ
Nb n; 2ÿj ; Card U n;w;r ; w 2 W  jˆp

hence the ®rst term in the right hand side of Eq. (17) is not larger than

CLT for `1 -valued semimartingales

P

481

q q X X U n;w;r  …ln ÿ mn † > sup 2ÿr fnr a; Xna;p;q t

n

…t;w†2S rˆp



q X

r Y

rˆp

jˆp

ÿ

Nb n; 2

ÿj




rˆp

!

ajfn j2 2 exp ÿ r 4

!

!  2ga=4 :

Consequently we obtain (I)  2ga=4 ‡ P n …snq < s†. Estimation of (II). To begin with, observe that Ln;w;p;q ÿ Ln;w;…p† ˆ

q X ÿ rˆp‡1

ˆ


W n;w;r ÿ W n;w;p 1Bnr …w†  …ln ÿ mn †

qÿ1 X rÿ1 ÿ X rˆp‡1 uˆp

‡

qÿ1 X ÿ uˆp

ˆ

qÿ1 X


W n;w;u‡1 ÿ W n;w;u 1Bnr …w†  …ln ÿ mn †


W n;w;u‡1 ÿ W n;w;u 1Bnq …w†  …ln ÿ mn †

Z n;w;u  …ln ÿ mn † ;

uˆp

ÿ
S where Z n;w;u ˆ W n;w;u‡1 ÿ W n;w;u 1Cun …w† with Cun …w† ˆ qrˆu‡1 Bnr …w†. Here, for every p  u  q ÿ 1, it follows from 0  W n;w;p;u‡1 ÿ W n;w;p;u  W

n;w;p;u

ÿ W n;w;p;u

that jZ n;w;u j2  mns  2ÿ2u a and that

on the set

Xna;p;q

ÿ
ÿ1 0  Z n;w;u  2u‡1 fnu‡1

[notice also Cun …w†  …Anu …w††c to get the latter fact]. So Corollary 3.3 (a) yields that   n;w;u n n ÿ…u‡1† n n  …l ÿ m †t > 2 fu‡1 a; Xa;p;q P sup Z n

t2S

 2 exp ÿ

j2ÿ…u‡1† fnu‡1 aj2

!

2‰…2u‡1 fnu‡1 †ÿ1
2ÿ…u‡1† fnu‡1 a ‡ 2ÿ2u aŠ ! ajfnu‡1 j2 ˆ 2 exp ÿ : 10

~ n n Su Bn …w†, it is clear that Since Cun …w† ˆ X rˆp r

482

Y. Nishiyama u‡1  Y ÿ
Card Z n;w;u ; w 2 W  Nb n; 2ÿj ; jˆp

hence we obtain (II)  P n? 

qÿ1 qÿ1 X X n;w;u Z sup  …ln ÿ mn †t > a 2ÿ…u‡1† fnu‡1 ; Xna;p;q

…t;w†2S uˆp

qÿ1 X

u‡1 Y

uˆp

jˆp

ÿ Nb n; 2

!
ÿj

!

uˆp

2 exp ÿ

ajfnu‡1 j2 10

!

 2ga=10 :

Estimation of …III†. Here we consider the case of S ˆ ‰0; sŠ only. The argument for the case of S ˆ fsg is easier. Notice that D

E Ln;w;…p† ÿ Ln;/;…p† ; Ln;w;…p† ÿ Ln;/;…p† t D E n;w;…p† n;w;…p† n;w;…p† 3 L ÿM ;L ÿ M n;w;…p† t D E n;w;…p† n;/;…p† n;w;…p† n;/;…p† ‡3 M ÿM ;M ÿM D E t n;/;…p† n;/;…p† n;/;…p† n;/;…p† ‡3 M ÿL ;M ÿL t ÿ n ÿ n;w n;/
2 n n;w;p n;w;p
2  3 q W ;W ‡ 3 q W ;W t t ÿ n;/;p
2 ‡ 3 qn W n;/;p ; W : t

For every pair w; / 2 W, recalling that the function t , q……t; w†; …t; /†† is continuous, we set rp …w; /† ˆ supft 2 S; q……t; w†; …t; /††  2ÿp g : Then it follows from Corollary 3.3 (a) that for every  > 0 ! n;w;…p† n;/;…p† n n? (III) ˆ P sup sup Lt ÿ Lt > ; Xa;p;q w;/2W t2‰0;rp …w;/†Š

  Nb …n; 2ÿp †2
2 exp ÿ ‡ P n?

 2 2‰…2
2ÿ2p †
 ‡ 9a
2ÿ2p Š ! ÿ 2 n n;w n;/
ÿ2p sup q W ; W >a
2 : rp …w;/†

w;/2W

These estimates of (I), (II) and (III) complete the proof.

(

Proof of Theorem 2.2. First notice that the condition …B1 † implies Pn

V n 1fV n >eg  mns ÿ! 0

8e > 0 :

…18†

CLT for `1 -valued semimartingales

483

Hereafter we shall check the conditions of Theorem 2.1. It is clear from Eq. (6) that the condition …A1 † is satis®ed (recall also Remark 2). Next, the canonical decomposition of X n;w ˆ W n;w  …ln ÿ mn † is given as follows: X0n;w ˆ 0 ; X n;w;a ˆ W n;w 1fjW n;w j>ag  ln ; Bn;w;a ˆ ÿW n;w 1fjW n;w j>ag  mn ; M n;w;a ˆ W n;w 1fjW n;w jag  …ln ÿ mn † : Now observe that for every e > 0 ! n;w;a n? Xt > e ˆ P n? sup P …t;w†2‰0;sŠW

sup

…t;w†2‰0;sŠW

! n;w n W 1fjW n;w j>ag  l > e t

ÿ
 P n V n 1fV n >ag  lns > e ÿ
 e ‡ P n V n 1fV n >ag  mns > e2 ;

where the last inequality follows from the Lenglart's inequality (e.g. Lemma I.3.30 of [22]). Hence Eq. (18) implies the conditions …A2 † and …A3 †. Also, under …B1 †, the condition …A4 † is deduced from …B2 †. Further, by virtue of Lemma 3.8, it is sucient for …A6 † to show that for every e; g > 0 and every p2N ! n;w;2ÿ2p n;w;…p† n? sup Mt ÿ Mt lim sup P …19† >e g : n!1

…t;w†2S

To do it, notice that n;w;2ÿ2p n;w;…p† ÿ Mt Mt n;w ˆ W 1fjW n;w j2ÿ2p g  …ln ÿ mn †t ÿW n;w 1fV n 2ÿ2p g  …ln ÿ mn †t ˆ W n;w 1fjW n;w j2ÿ2p
 V n 1fV n >2ÿ2p g  lnt ‡ V n 1fV n >2ÿ2p g  mnt : So the Lenglart's inequality and Eq. (18) yield Eq. (19). The proof is ®nished. (

3.4. Proof of Theorem 2.3. Let us denote by Ns …e† ˆ N …W; qs ; e† the e-covering number of W with respect to the metric qs de®ned by qs …w; /† ˆ q……s; w†; …s; /††. Notice that Ns …e†  N …S; q; e†. We introduce the following notations, which are similar to those in Sect. 3.3: for every p; r 2 N and a  1

484

Y. Nishiyama

p fr ˆ log 2r Ns …2ÿr † ; 1 X 2ÿr fr ; ep ˆ rˆp

ga ˆ

1 X rˆ1

  Ns …2ÿr † exp ÿajfr j2 :

Then the following facts are derived from the same argument as for Lemma 3.6 (in fact, the proofs are easier): under the condition …C3 †, lim ga ˆ 0

a!1

and

lim ep ˆ 0 :

…20†

p!1

In order to prove Theorem 2.3. we shall check the conditions in Theorem 2.1. The direct proof is also possible, but it is essentially same as the one presented here. The chaining argument below is taken from Theorem 11.6 of [24]. Proof of Theorem 2.3. By virtue of Theorem 2.1, it is enough to show that 0 lim lim sup P n @

p!1

n!1

sup

t2S; w;/2Wn 0
1 n;w Xt ÿ Xtn;/ > eA ˆ 0

8e > 0 :

…21†

To do it, ®x any n 2 N for a while. Let fWnm gm2N be a sequence of ®nite subsets of Wn such that Wnm " Wn as m ! 1. For every m; p 2 N let us denote by q…n; m; p† the smallest integer such that q…n; m; p† > p and that each of closed balls with centers in Wnm and radius 2
2ÿq…n;m;p† in the proper metric qs contains exactly one point in Wnm . Then we have Card…Wnm †  Ns …2ÿq…n;m;p† †. of Wnm , p  r < q…n; m; p†, such that: Next we choose some subsets Wn;m;p r …i† Card…Wn;m;p †  Ns …2ÿr †; r : Wnm ! Wn;m;p such that qs …w; krn;m;p …w†† …ii† there exists a mapping kn;m;p r r n ÿr  2
2 for every w 2 Wm . n;m;p n n Set Wn;m;p q…n;m;p† ˆ Wm , and denote by kq…n;m;p† the identical mapping on Wm . Now, n;m;p n n;m;p : Wm ! Wr , p  r  q…n; m; p†, by we de®ne the mappings wr n;m;p ˆ kn;m;p  kn;m;p wn;m;p r r r‡1 


 kq…n;m;p† :

Further, for given a > 0 let us de®ne the set Xna;p 2 Fns by Xna;p ˆ Since

8 < :

sup

t2S; w;/2Wn 0
9 p= cn …w; /†t  a : ; q……t; w†; …t; /†† ÿp

CLT for `1 -valued semimartingales

485

q…n;m;p† X n;wn;m;p …w† n;wn;m;p …w† n;w n;/ ÿ Xt rÿ1 Xt ÿ Xt  Xt r rˆp‡1

‡

q…n;m;p† X rˆp‡1

n;wn;m;p …/† n;wn;m;p …/† ÿ Xt rÿ1 Xt r

n;wn;m;p …/† n;wn;m;p …w† ÿ Xt p ‡ Xt p ; it holds that for every  > 0 0 B P n@

sup

t2S; w;/2Wnm 0
1 n;w C n;/ Xt ÿ Xt > 2aep‡1 ‡ A

   P n Xn n Xna;p ‡ (I) ‡ (II) ; where the terms (I) and (II) in the right hand side are given by: (I) ˆ P

n

q…n;m;p† X

sup

t2S; w2Wnm rˆp‡1

0 B (II) ˆ P n @

sup

t2S; w;/2Wnm 0
n;wn;m;p …w† n;wn;m;p …w† ÿ Xt rÿ1 > aep‡1 ; Xna;p Xt r

…22† ! ;

1 …w† n;wn;m;p …/† n;wn;m;p C ÿ Xt p Xt p > ; Xna;p A :

Hereafter we shall estimate (I) and (II). Estimation of (I). It follows from Lemma 3.2 (ii) that (I) 

q…n;m;p† X

X

rˆp‡1 w2Wn;m;p r

  n;kn;m;p …w† P n sup Xtn;w ÿ Xt rÿ1 > 2ÿr fr
a; Xna;p t2S

! j2ÿr fr
aj2 Ns …2 †
2 exp ÿ  2
2ÿ2r‡4 a rˆp‡1 ! q…n;m;p† X ajfr j2 ÿr Ns …2 †
2 exp ÿ   2ga=32 : 32 rˆp‡1 q…n;m;p† X

ÿr

Estimation of (II). Here we consider the case of S ˆ ‰0; sŠ only. The argument for the case of S ˆ fsg is easier. Recall (iii) in the condition …M†: the functions t , q……t; w†; …t; /†† are assumed to be continuous and nondecreasing. For every t 2 S; w; / 2 Wnm such that q……t; w†; …t; /††  2ÿp , we have X ÿÿ
ÿ

q…n;m;p†
ÿ n;m;p

ÿÿ …w† ; t; wpn;m;p …/†  q t; wn;m;p …w† ; t; wrÿ1 …w† q t; wn;m;p p r rˆp‡1

486

Y. Nishiyama

‡

q…n;m;p† X rˆp‡1

ÿÿ
ÿ

q t; wrn;m;p …/† ; t; wn;m;p ‡ q……t; w†; …t; /†† rÿ1 …/†

 9
2ÿp : So, for every pair w; / 2 Wpn;m;p we set …w; /† ˆ supft 2 S; q……t; w†; …t; /††  9
2ÿp g : rn;m;p p Then it follows from Lemma 3.2 (ii) that ! X n;w n;/ n n (II)  P sup Xt ÿ Xt > ; Xa;p w;/2Wn;m;p p

t2‰0;rn;m;p …w;/†Š p



2  Ns …2 †
2 exp ÿ 2
81a
2ÿ2p ÿp 2

 :

From these estimates of (I) and (II), by letting m ! 1 in the left hand side of Eq. (22) we obtain 0 P n@

sup

t2S; w;/2Wn 0
1 n;w n;/ Xt ÿ Xt > 2aep‡1 ‡ A

   P n Xn n Xna;p ‡ 2ga=32    2 2p ÿ2p‡1 ÿp : ‡ 2 exp 2 2 log Ns …2 † ÿ 162a Therefore we can deduce Eq. (21) by using Eq. (20): ®rst choose a large ( constant a, and then, take limp!1 lim supn!1 .

4. Nonparametric estimation for marked point processes In this section we concentrate on the case of S ˆ fsg for simplicity. Let …E; E† be a Blackwell space on which a r-®nite measure k is de®ned. For every n 2 N, let ln be an integer-valued random measure on R‡  E, de®ned on a ®ltered probability space …Xn ; Fn ; Fn ; P n †. Notice that ln can be identi®ed with an E-marked point process f…Tin ; Zin †; i 2 Ng through the equality X e…Tin …x†;Zin …x†† …dt; dz† ; ln …x; dt; dz† ˆ i

T1n

T2n

< <


a.s. and each Zin is an E-valued random variable. where 0 < We assume that the predictable compensator mn of ln is given by mn …x; dt; dz† ˆ a…t; z†Y n …x; t; z†dtk…dz† ;

CLT for `1 -valued semimartingales

487

where a…t; z† is a non-negative real-valued measurable function on R‡  E, and Y n …x; t; z† is a non-negative real-valued predictable function on Xn  R‡  E. Further, for some ®xed constant s > 0, we also assume Z a…t; z†dtk…dz† < 1 ‰0;sŠE

and sup

…t;z†2‰0;sŠE

Y n …x; t; z† < 1

8x 2 Xn :

We will consider the estimation problems for unknown function a, based on the information of Fns (hence, in addition to ln , the predictable function Y n is also assumed to be observable on ‰0; sŠ  E). Here we make the following condition, which is a natural generalization of (8.4.1) in [3]. Condition 4.1. There exist a measurable function y ˆ y…t; z† on ‰0; sŠ  E, which is bounded and bounded away from zero, and a sequence fan g of positive constants such that an " 1 and that sup

…t;z†2‰0;sŠE

ÿ2 n P n? a Y …
; t; z† ÿ y…t; z† ÿ !0 : n

Based on the function y appeared above, we set H ˆ L2 …‰0; sŠ  E; a=y† and equip it with the inner product Z a…t; z† dtk…dz† : …h1 ; h2 †H ˆ h1 …t; z†h2 …t; z† y…t; z† ‰0;sŠE p Let us de®ne the pseudo-metric q on H by q…h1 ; h2 † ˆ …h1 ÿ h2 ; h1 ÿ h2 †H . Now we ®x a subset W  H which satis®es the following condition. Condition 4.2. The set W  H has the envelope u 2 H, that is jw…t; z†j  u…t; z† 8…t; z† 2 ‰0; sŠ  E 8w 2 W : Further …W; q† satis®es the metric entropy condition with bracketing: Z 1 p log Nb …W; q; e† de < 1 : 0

1

We equip ` …W† with the sup-norm jj
jj1 . We can consider A ˆ …A…w†jw 2 W† and F ˆ …F …w†jw 2 W†, which are given by Eq. (2) and Eq. (3) respectively, as `1 …W†-valued unknown parameters. First, in order to estimate the (weighted) cumulative hazard function A ˆ …A…w†jw 2 W†, we introduce the generalized Nelson-Aalen's estimator A^n ˆ …A^n …w†jw 2 W† de®ned by Z n ^ w…t; z†Y nÿ …x; t; z†ln …x; dt; dz† A …w† ˆ ‰0;sŠE

ˆ …wY nÿ †  lns ;

488

Y. Nishiyama

where Y nÿ …x; t; z† ˆ



1 Y n …x;t;z†

if Y n …x; t; z† > 0 ; if Y n …x; t; z† ˆ 0 :

0

Theorem 4.3. Under Condition 4.1 and 4.2, it holds that   Pn an A^n ÿ A ˆ) G

in

`1 …W† ;

where the Borel probability measure G on …`1 …W†; jj
jj1 † is concentrated on the set Uq …W†, and its ®nite-dimensional projection Gw1 ;...;wd is the centered Gaussian distribution with covariance matrix R ˆ fRij g given by Rij ˆ …wi ; wj †H . Proof. We shall apply Theorem 2.2 in the following way: W n;w ˆ an wY nÿ n

V ˆ an uY

nÿ

8w 2 W ; :

The conditions …B1 † and …B2 † are straight from jV n j2 1fV n >eg  mns Z 2 ˆ an ju…t; z†j2 Y nÿ …t; z†1fjan u…t;z†Y nÿ …t;z†j>eg a…t; z† dtk…dz† ‰0;sŠE

and W

n;w

W

n;/



mns

a2n

ˆ

Z ‰0;sŠE

w…t; z†/…t; z†Y nÿ …t; z†a…t; z† dtk…dz† ;

respectively. Next, …B3 † follows from Condition 4.2. Further, under Condition 4.1, there exists a constant L > 0 such that if we set ( XnL

ˆ

x;

sup

a2n Y nÿ …x; t; z†

…t;z†2‰0;sŠE

y…t; z†ÿ1

) L

then limn!1 P n? …Xn n XnL † ˆ 0. Since Z n n;w n;/ 2 q …W ; W † ˆ a2 jw…t; z† ÿ /…t; z†j2 Y nÿ …t; z†a…t; z†dtk…dz† n s Z L

‰0;sŠE

‰0;sŠE

jw…t; z† ÿ /…t; z†j2

a…t; z† dtk…dz† on the set XnL ; y…t; z†

we obtain …B4 †. In order to check …B5 †, for every n 2 N and e > 0 we in~ n -measurable bracket f…W n;e;l;k ; W n;e;l;k †; 1  k  Nb …W; q; e†g troduce the P n for W given by W n;e;l;k ˆ an we;l;k Y nÿ

and

W n;e;u;k ˆ an we;u;k Y nÿ ;

CLT for `1 -valued semimartingales

489

where f…we;l;k ; we;u;k †; 1  k  Nb …W; q; e†g is the B‰0; sŠ E-measurable bracket for W corresponding to Condition 4.2. The same argument as for ( …B4 † yields …B5 †. Secondly, in order to estimate the (weighted) distribution function F ˆ …F …w†jw 2 W† given by Eq. (3), we introduce the estimator F^n ˆ …F^n …w†jw 2 W† de®ned by ÿ
F^n …w† ˆ wY nÿ Kÿn  lns ; Q where Ktn ˆ st ‰1 ÿ D…Y nÿ  lns †Š. Corollary 4.4. Under Condition 4.1 and 4.2, it holds that   Pn an F^n ÿ F ˆ) G

in

`1 …W† ;

where the Borel probability measure G on …`1 …W†; jj
jj1 † is concentrated on the set Uq …W†, and its ®nite-dimensional projection Gw1 ;...;wd is the centered Gaussian distribution with covariance matrix R ˆ fRij g given by ÿ
R wi …t; z†wj …t; z†a…t; z† exp ÿ 2 ‰0;tŠE a…s; x†dsk…dx†

Z Rij ˆ

y…t; z†

‰0;sŠE

dtk…dz† :

Proof. It is well known that ! Z n n P sup Kt ÿ exp ÿ a…s; x†dsk…dx† ÿ! 0 t2‰0;sŠ ‰0;tŠE (see e.g. Theorem IV.3.1 of [3]). Hence we obtain the assertion by the same argument as for Theorem 4.3. ( Finally, let us discuss the asymptotic eciency of proposed estimators. According to Sect. VIII.3.1 of [3], we shall check the local asymptotic normality of the sequence of statistical experiments, and the di€erentiability of unknown parameters. See [7], Ibragimov and Khas'minskii [20] and van der Vaart [33] for more details about the asymptotic eciency theory in in®nitedimensional statistical models. [Local Asymptotic Normality] Let us ®x a convex cone C  H such that for every h 2 C there exists a constant l 2 R such that h  l. Let Pn ˆ fPhn ; h 2 Cg be a family of probability measures on the ®ltered space …Xn ; Fn ; Fn † such that: the predictable compensator mn;h of ln with respect to Phn is given by mn;h …x; dt; dz† ˆ anh …t; z†Y n …x; t; z†dtk…dz† ; where anh ˆ af1 ‡ ahn y g. Here we make the following condition.

490

Y. Nishiyama

T Condition 4.5. The ®ltration Fn ˆ …Fnt †t2R‡ satis®es Fnt ˆ s>t Hns with Hns ˆ Fn0 _ r…ln …‰0; rŠ  U † : r  s; U 2 E†. Also, for every h 2 C it holds that Phn ˆ P0n on Fn0 and that Phn  P0n on Fns . Under the preceding condition, the log-likelihood ratio is given by    dP nh jFns h h n  lns ÿ  mn;0 K …h† ˆ log n n ˆ log 1 ‡ dP 0 jFs an y an y s

…23†

(see e.g. Theorem III.5.43 of [22]). Hence, it follows from the usual semimartingale central limit theorem that, under Condition 4.1. and 4.5, the sequence of statistical experiments …Xn ; Fns ; Pn ˆ fPhn ; h 2 Cg† is locally asymptotic normal; that is, for every d 2 N and h1 ; . . . ; hd 2 C P0n ÿ
…Kn …h1 †; . . . ; Kn …hd †† ˆ) N ÿ12 diagR; R

in

Rd

where Rij ˆ …hi ; hj †H . [Di€erentiability of Parameters] We prepare some notations. We set …B; jj
jjB † ˆ …`1 …W†; jj
jj1 †, and denote by B the dual space of B. For given T : H ! B and g 2 B , we de®ne Tg by the unique element in H such that …h; Tg †H ˆ g  T …h† for every h 2 H. We denote by pw , which belongs to B , the projection to w. The parameter A. Let us de®ne the mapping An : H ! B by n A …h† ˆ …An …h†…w†jw 2 W† where Z n w…t; z†anh …t; z†dtk…dz† : A …h†…w† ˆ ‰0;sŠE

Then, it is trivial that An is di€erentiable with rate an and its derivative A0 : H ! B is given by A0 …h† ˆ …A0 …h†…w†jw 2 W† where Z a…t; z† A0 …h†…w† ˆ dtk…dz† : w…t; z†h…t; z† y…t; z† ‰0;sŠE Further, we have A0pw ˆ w for every w 2 W. The parameter F . Let us de®ne the mapping F n : H ! B by n F …h† ˆ …F n …h†…w†jw 2 W† where ! Z Z n n n w…t; z†ah …t; z† exp ÿ ah …s; x†dsk…dx† dtk…dz† : F …h†…w† ˆ ‰0;sŠE

‰0;tŠE

Then, it is trivial that F n is di€erentiable with rate an and its derivative F 0 : H ! B is given by F 0 …h† ˆ …F 0 …h†…w†jw 2 W† where ! Z Z a…t; z† 0 exp ÿ F …h†…w† ˆ w…t; z†h…t; z† a…s; x†dsk…dx† dtk…dz† : y…t; z† ‰0;sŠE ‰0;tŠE R Further, we have Fp0 w ˆ Fp0 w …t; z† ˆ w…t; z† exp…ÿ ‰0;tŠE a…s; x†dsk…dx†† for every w 2 W. From the discussion above, we can conclude:

CLT for `1 -valued semimartingales

491

Corollary 4.6. Under Condition 4.1, 4.2 and 4.5, if the set W is included by the closed linear span of C, then the estimators A^n for A and F^n for F are respectively asymptotically ecient in the sense of convolution theorem. 5. Log-likelihood ratio random ®elds of continuous semimartingales For every n 2 N, let …Xn ; Fn ; Fn † be a ®ltered space, on which a real-valued continuous adapted process X n is de®ned. Let C be a family of locally bounded measurable functions on R‡ such that 0 ˆ 0…t† 2 C. Let fan g be a sequence of positive constants such that an " 1. Let Pn ˆ fPhn ; h 2 Cg be a family of probability measures on the ®ltered space …Xn ; Fn ; Fn † such that: under Phn , the process X n is a semimartingale with the local characteristics …Bn;h ; C n;h ; 0† given by Z t ˆ anh …s†Zsn Ysn ds with anh …t† ˆ a…t† ‡ aÿ1 Bn;h t n h…t† ; 0 Z t Ctn;h ˆ Ysn ds ; 0

where Y n and Z n are predictable processes. In other words, under Phn , the process X n has the semimartingale decomposition dX nt ˆ anh …t†Ztn Ytn dt ‡ dM n;h t

Rt where Mtn;h is a continuous local martingale with hM n;h ; M n;h it ˆ 0 Ysn ds. It is well known that, under some mild conditions, the log-likelihood ratio process t , Kn …h†t is given by dP nh jFnt dP n0 jFnt Z 2 ÿ
1 t ÿ1 n ˆÿ an h…s†Zsn Ysn ds ‡ aÿ1
Mtn;0 n hZ 2 0

Kn …h†t ˆ log

…24†

(see e.g. Theorem III.5.34 of [22]). Here we make the following condition. Condition 5.1. There exists an integrable function g ˆ g…t† on ‰0; sŠ, which is bounded away from zero, such that ÿ Pn
0 n 2 n Yt ÿ g…t† ÿ! 0 : sup aÿ1 n Zt

t2‰0;sŠ

2 Using the function g appeared above, we R s set H ˆ L …‰0; sŠ; g…t†dt† and equip h1 …t†h2 …t†g…t†dt. Let it with the inner product …h1 ; h2 †H ˆ p 0   us de®ne the pseudo-metric . on H by .…h1 ; h2 † ˆ …h1 ÿ h2 ; h1 ÿ h2 †H . Now we ®x a subset W  C which satis®es the following condition.

Condition 5.2. The set W  C has the envelope u 2 H, that is

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jh…t†j  u…t†

8t 2 ‰0; sŠ

8h 2 W :

Also . is reduced to the proper metric on W. Further …W; .† satis®es the metric entropy condition: Z 1 p log N …W; .; e† de < 1 : 0

We de®ne the pseudo-metric q on S ˆ ‰0; sŠ  W by  s Z q……t1 ; h1 †; …t2 ; h2 †† ˆ

t1 _t2

t1 ^t2

ju…s†j2 g…s†ds ‡ …h1 ÿ h2 ; h1 ÿ h2 †H :

Theorem 5.3. Suppose that each log-likelihood ratio process t , Kn …h†t is given by the right hand side of Eq. (24), and that Condition 5.1 and 5.2 are satis®ed. Suppose also that Kn ˆ …Kn …h†t j…t; h† 2 S† is separable with respect to the pseudo-metric q under P0n , and that a version of Kn takes values in `1 …S†. Then, it holds that P0n

Kn ˆ) G

in

`1 …S† ;

where the Borel probability measure G on …`1 …S†; jj
jj1 † is concentrated on the set Uq …S†, and its ®nite-dimensional projection R t ^t G…t1 ;h1 †;...;…td ;hd † is the Gaussian distribution N …ÿ12 diag R; R† where Rij ˆ 0i j hi …s†hj …s†g…s†ds. Proof. Notice that for every t 2 ‰0; sŠ and h1 ; h2 2 W

ÿ ÿ1
ÿ
 n an h1 Z n
M n;0 ; aÿ1
M n;0 t ˆ n h2 Z

Z

t 0

n 2 n h1 …s†h2 …s† aÿ1 n Zs Ys ds :

We can easily obtain the assertion from Theorem 2.3 (repeat the same argument as in the proof of Theorem 4.3). ( Next, ®x a subset W1  C which satis®es Condition 5.2 and jjhjjH ˆ 1 for every h 2 W1 . Let us consider testing hypothesis H 0 : a ˆ a0 against H1 : a ˆ a0 ‡ aÿ1 n h; h 2 W1 : We introduce the test statistics Tn ˆ

1 ‡ sup Kn …h†s : 2 h2W1

Then, it directly follows from Theorem 5.3 that: Corollary 5.4. Under the same conditions as in Theorem 5.3, it holds that P0n

T n ˆ) L

in R ;

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493

~ where L denotes the distribution of random variable suph2W1 K…h†, and where ~ ~ K ˆ …K…h†jh 2 W1 † is a U. …W1 †-valued random element whose ®nite-dimen~ d ††  N …0; R† with ~ 1 †; . . . ; K…h sional distributions are given by: …K…h Rij ˆ …hi ; hj †H . 6. Final remarks It is possible to derive the asymptotic behavior of log-likelihood ratio random ®elds of marked point processes, given by Eq. (23), in the same way as Theorem 5.3. R s Conversely, we can also construct an estimator for A ˆ A…w† ˆ 0 w…t†a…t†dt in Sect. 5, and show its functional asymptotic normality and asymptotic eciency as in Sect. 4. Greenwood and Wefelmeyer [15], [16], [17], [18] have successfully taken such an approach to estimate some ®nite-dimensional parameters in nonparametric models for stochastic processes (i.e. the function w in our notations is ®xed). Further they suggested the possibility of generalization to the cases of in®nite-dimensional parameters (i.e. the function w runs through W), referring the work [5] (see 9p. of [16] and 138p. of [18]). The crucial point in those problems is to show the tightness of proposed estimators, and our central limit theorems in Sect. 2 are useful to solve it. Hopefully, the results in Sects. 4 and 5 will be the prototypes of other various applications. Acknowledgements. I am grateful to Dr. S. Aki for his lectures on empirical processes in the winter of 1993±1994 at Osaka University, which inspired me to do this work. My thanks also go to Prof. J. Jacod and Prof. N. Yoshida for their valuable comments in several stages of the project.

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