Ukrainian Mathematical Journal, Vol. 63, No. 8, January, 2012 (Ukrainian Original Vol. 63, No. 8, August, 2011)
SOJOURN TIME OF ALMOST SEMICONTINUOUS INTEGRAL-VALUED PROCESSES IN A FIXED STATE D. V. Gusak
UDC 519.21
Let .t / be an almost lower semicontinuous integer-valued process with moment generating function of the negative parts of jumps k W EŒz k =k < 0 D
1 z
b ; b
0 b < 1:
For the moment generating function of the sojourn time of .t/ in a fixed state, we deduce relations in terms of the roots zs < 1 < zys of the Lundberg equation. Passing to the limit as s ! 0 in the relations obtained as a result, we determine the distributions of lr .1/:
The sojourn times of the processes with independent increments and values in R were studied in [1, 2]. For integer-valued additive sequences and semi-Markov processes, the distributions of sojourn times were investigated and the relationship for the generatrix of sojourn times was established in [3–5]. The same relations were deduced in [5] for any integer-valued sojourn time without application of the factorization. The relationships for the distribution of sojourn times of almost continuous integer-valued processes .t / in any state are specified in [5] on the basis of the results of correction of the representations for the components of factorization [see Theorems 7.5 and 7.6 in the cited monograph]. For this purpose, we use the relations for the distribution of .t / expressed via the roots of the Lundberg equation zs < 1 < zys ; s > 0: The integer-valued Poisson process .t / ..0/ D 0; t 0/ is called almost lower semicontinuous if it crosses a negative level x < 0 only with the help of negative geometrically distributed jumps k < 0 W p.z/ D EŒz k j k < 0 D
1 z
b ; b
0 b < 1:
For b D 0; the process .t / is called lower semicontinuous. Let .t / be an almost lower semicontinuous integer-valued process with cumulant k.z/ D ln Ez
.t/
1 D pp.1/ .z/ C q z
b b
1 ;
jzj D 1;
p C q D 1;
(1)
where p.1/ .z/ D EŒz 1 j1 > 0; b 2 .0; 1/ is a parameter of the geometric distribution of k < 0; and > 0 is the intensity of jumps k : By s we denote a random variable with exponential distribution: Pfs > t g D e st ; s > 0: We also introduce the following notation: Zt lr .t / D
I f.u/ D rgdu;
C .z/ D infft > 0W .t/ > rg;
r 2 Z;
0
Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 8, pp. 1021–1029, August, 2011. Original article submitted November 18, 2010; revision submitted June 23, 2011. 1176
0041-5995/12/6308–1176
c 2012
Springer Science+Business Media, Inc.
S OJOURN T IME OF A LMOST S EMICONTINUOUS I NTEGRAL -VALUED P ROCESSES IN A F IXED S TATE
˙ .t / D sup.inf/ .t 0 /;
˙ D sup.inf/ .t/;
0t 0 t
dr .s; / D Ee
lr .s /
1177
0<1
Z1 Ds
st
e
Ee
lr .t /
dt;
0
g.s; z/ D Ez
.s /
Z1 Ds
Ez .t / dt D s.s
k.z//
1
;
0
g˙ .s; z/ D Ez
˙ .s /
Z1 Ds
Ez
˙ .t /
dt:
0
In [5], it is also shown that, for .t / with cumulant (1), the Lundberg equation ( k.z/ D s ) has two prime roots: b < zs < 1 < zys : For m D E.1/ D 0 and s ! 0; these roots converge to 1. Thus, for m > 0; we have zs ! z0 < 1 and
zys ! 1:
zys ! zy0 > 1 and
zs ! 1:
s!0
s!0
At the same time, for m < 0; we have s!0
s!0
Moreover, zs D q .s/ C bp .s/ < 1;
p .s/ D Pf .s / D 0g;
q .s/ D 1
and
p .s/:
In the case of almost lower semicontinuity for the main factorization identity g.s; z/ D gC .s; z/g .s; z/;
jzj D 1;
(2)
the component g .s; z/ is completely determined by the smaller root of the Lundberg equation k.z/ D s; i.e., zs 2 .b; 1/ (see Theorem 7.6 in [5]): g .s; z/ D
p .s/.z b/ ; z zs
p .s/ D Pf .s / D 0g D
1 1
zs : b
(3)
According to (7.32) in [5], the generatrix gC .s; z/ can be rewritten in a somewhat simpler form than in the Spitzer formula, namely, gC .s; z/ D
1 b g0 .s; z/ C q .s/ p .s/ b
1 .g1 .s; b/ z
g1 .s; z// ; (4)
gk .s; z/ D
X rk
z r pr .s/;
pk .s/ D Pf.s / D kg;
k 0:
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We now consider an almost lower semicontinuous integer-valued risk process .t/ with the Erlang distribution of demands k > 0 p.1/ .z/ D z n
Y 1 cr ; 1 cr z rn
0 cr < 1;
and the geometric distribution of “bonuses” k < 0 (see p.z/ with b 2 Œ0; 1/ mentioned above) and prove the following proposition: Lemma 0. For the risk process with the Erlang distribution of demands, the generatrix .s / admits the following representation: s.z b/qn .z/ ; k .s; z/
g.s; z/ D
k .s; z/ D s.z
qn .z/ D
Y
.1
q1 .z/ D 1
cr z/;
cz;
(5)
rn
b/qn .z/
Œp.z
b/z n qn .1/ C pb C q
z:
(6)
In view of the existence of the roots zs < 1 < zys of the Lundberg equation ( k.z/ D s k .s; z/ D 0 ), k .s; z/ D PnC1 .s; z/ D .y zs
z/.z
zs /Pn
1 .s; z/;
(7)
where Pn .s; z/ is a polynomial of the nth order. If some constants cr D 0; then qm .z/ is a polynomial of order m < n: The generatrix C .s / can be represented in the form gC .s; z/ D s p .s/Q.s; z/.y zs
z/
1
;
(8)
where Q.s; z/ D Pn
1
n 0;
gC .s; z/ ! pC .s/ D
In particular, if only c1 D c ¤ 0 and q1 .z/ D 1
cz (i.e., for n D 1/; then
1 .s; z/qn
gC .s; z/ D
z!0
s 1 cz ; p .s/ .y zs z/Q0 .s/
agrees with relation (7.27) in [5]
.z/;
s : Q.s; 0/y zs p .s/
zys D .pC .s/ C cqC .s//
pC .s/.1 cz/y zs gC .s; z/ D zys z
1
;
:
As a result of the substitution of p1 .z/ in (1) and simple transformations, we establish (5) for s and (6) for kC .s; z/: Further, substituting (3) and (5) in relation (2) with regard for (7), s k.z/ we arrive at the relation Proof. g.s; z/ D
.z which yields (8) for gC .s; z/:
.z b/qn .z/s p .s/.z b/ D gC .s; z/; zs /.y zs z/Pn 1 .s; z/ z zs
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In the general case, we have g.s; z/ D
s.z b/ ; k .s; z/
k .s; z/ D s.z
b/
h p.z
b/p.1/ .z/ C pb C q
z
i
due to the existence of the roots of the Lundberg equation zs < 1 < zys : k.z/ D s k .s; z/ D 0;
k .s; z/ D Q.s; z/.y zs
z/.z
zs /:
Instead of (5), we obtain g.s; z/ D
.y zs
s.z b/ ; z/.z zs /Q.s; z/
Q.s; zs /Q.s; zys / ¤ 0:
Similarly, by using (2), (3), and the last representation for g.s; z/; we arrive at the relation
.z
.z b/s z D zs /.y zs z/Q.s; z/ z
b p .s/gC .s; z/; zs
which yields formula (8). In the general case, we can find only partial values of Q.s; z/: In particular, it follows from relation (8) with z D 1 and z D zs < 1 that Q.s; 1/.y zs
1/ D sp .s/
1
gC .s; zs / D s p .s/Q.s; zs /.y zs
;
zs /
1
:
(9)
For m < 0; zs ! 1; and zys ! z0 > 1; it follows from relation (9) that Q.s; zs /.y zs
zs / ! Q.0; 1/.y z0 s!0
1/ D jmj.1
b/:
(10)
As shown in what follows, despite the fact that the complete information about Q.s; z/ is absent, in the limiting case as s ! 0; we get the relation for the distribution lr .1/ by using (8) with m 0: We now present the main relations from Theorem 7.8 in [5] in the general case. They should be corrected for .t/ with cumulant (1) in the case of almost lower semicontinuity. Theorem 1 [5]. Any integer-valued process .t/ satisfies the following relations for the generatrices lr .s / W d0 .s; / D
s ; s C p0 .s/
dr .s; / D 1
pr .s/ D Pf.s / D rg
pr .s/ ; s C p0 .s/
As a result of the inversion with respect to ; relation (11) yields
r ¤ 0:
for any r;
(11)
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Pfl0 .s / > yg D e
ysp0 1 .s/
y 0;
;
pr .s/p0 1 .s/;
Pflr .s / D 0g D 1
ysp0 1 .s/
Pflr .s / > yg D pr .s/p0 1 .s/e
r ¤ 0;
! pr .s/p0 1 .s/;
(12)
y!0
Zt Pflr .t / > yg D
Pflr .t
u/ > 0gdu Pfl0 .u/ > yg;
r ¤ 0:
0
In order to correct (11) and (12) for .t / with cumulant (1), we use the lemma on the representation of pk .s/ in terms of zs D q .s/ C bq .s/ 2 .b; 1/: Lemma 1. Let .t / be an almost lower semicontinuous process with cumulant (1). The relation for the distributions ˙ .s / (see Theorem 7.6 in [5]) p
k .s/
b/zsk
D p .s/.zs
1
k 1;
;
p .s/ D
1 1
zs ; b (13)
pkC .s/ D
1 pk .s/ C .b p .s/
zs /b
k 1
gkC1 .s; b/ ;
p0˙ .s/ D p˙ .s/;
k > 0;
and the basic factorization identity (2) yield the following relations for pk .s/ expressed in terms of zs W p0 .s/ D p .s/ bzs 1 pC .s/ C .1 pk .s/ D p .s/ pkC .s/ C .1 p
gkC .s; z/ D
X
C zs 1 b/zs k gkC1 .s; zs / ;
b/zsk
k .s/ D p .s/.zs
prC .s/z r
zs 1 b/gC .s; zs / ;
1
k > 0; (14)
gC .s; zs /;
rD
k < 0;
g0C .s; z/ D gC .s; z/:
.k 0/;
rk
For the atomic probabilities p˙ .s/; the following relations are true in terms of zs : pC .s/ D
1 1
bh p0 .s/ C .b zs
pC .s/ D p0 .s/ p .s/ D .1
zs /b
p1 .s/zs .1
zs /.1
b/
1
1
i E b .s / ; .s / 1 ;
zs /
!1
1
for
b D 0;
zs
as
b ! 0:
(15)
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1181
Proof. Relations (14) are proved by expanding the generatrices of .s / and ˙ .s / in the Laurent series g.s; z/ D
1 X
z k pk .s/
and
g˙ .s; z/ D
kD 1
X
˙ z ˙k p˙k .s/;
k0
substituting these relations in (2), and multiplying the last two series by gathering the terms with the same powers of z: Relations (15) for pC .s/ and p .s/ follow from relations (4) and (3), respectively. In order to find the limits pk .s/ pzk D lim D s!0 s
Z1 Pf.t/ D kgdt;
k D 0; ˙1; : : : ;
0
whose values depend on the sign of m; we need the following lemmas (for the cases m D 0 and m ¤ 0 ). Lemma 2. Let .t / be an almost lower semicontinuous process with cumulant (1). Then, according to (8) and (14), the inverse value of the exponent of distribution (12) is given by the formula p0 .s/s If m D 0 .zs ! 1
1
D zs
b C zys Q.s; 0/ .y zs
zs b : zs /Q.s; zs /
(16)
0; zys ! 1 C 0 as s ! 0/; then lim
s!0
Further, if r D
1
p0 .s/ D pz0 D 1; s
1 D 0: pz0
(17)
k < 0 and m D 0; then, according to (8), p
pz
k
k .s/
D lim
p
s!0
D
s.zs b/zsk ; Q.s; zs /.y zs zs /
k .s/
s
p k .s/ D lim s!0 p0 .s/ s!0 zs lim
D 1;
1 D 0; pz k
.zs b/zsk b C bQ.s; zs /.y zs
(18)
zs /
D 1:
(19)
Moreover, if k > 0; then, according to (8) and (14), pk .s/ D
s Q.s; zs /.y zs
zs /
2 4.zs
b/zs k
1
C Q.s; zs /.y zs
zs /p .s/.pC .s/
.zs
b/zs k
1
k X rD0
3 zsr prC .s//5 :
(20)
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Finally, if m D 0; then, by analogy with the case k > 0; pk .s/ D 1; s!0 s
1 D 0; pzk
pzk D lim
k > 0;
(21)
as s ! 0: However, since pk .s/ p0 .s/ as s ! 0; the following relation is true: pk .s/ .1 D s!0 p0 .s/ .1 lim
b/Q.0; 1/ D 1; b/Q.0; 1/
k > 0:
Proof. The proof of Lemma 2 for m D 0 is, in fact, based on representation (8) for gC .s; zs / with singularity as s ! 0 .zs ; zys ! 1 0/: In order to apply relation (14) for pk .s/ with k > 0; it is necessary to represent this probability in terms of gC .s; zs / W 2 pk .s/ D p .s/ 4pkC .s/ C .zs
b/zs k
1
.gC .s; zs /
k X
3 prC .s/zsr /5 :
(22)
rD0
Substituting (8) in this relation, we get (20). Remark 1. According to Theorem 7.6 in [5], pk .s/ can be represented in terms of zs (see (13)). Multiplying the series 2
3
4p .s/ C
X
z k .zs
b/zsk
15
k1
X
z k pkC .s/
k0
with regard for (8) and (22), we arrive at relation (20) in terms of zs < 1 < zys and Q.s; zs /: Denote H.k/ D E C .k/; k > 0; hk D H.k/
H.0/ D E C .0/ D
H.k Z1
1/; and
Pf C .t/ D 0gdt D pPC :
0
We now prove a similar lemma for m ¤ 0 by denoting the limits as pzk D pzk> .pzk< / for ˙m > 0: Lemma 3. If m > 0; then p .s/ ! p ; s!0
zs ! z0 < 1; s!0
and there exist the limits lim s
s!0
1 C pr .s/
D pPrC D hr ;
r > 0;
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1183
via which the quantities pzk D pzk> are expressed in terms of z0 : 2
3
pz0> D p 4bz0 1 H.0/ C .1
z0 1 b/
X
z0r hr 5 ;
k D 0;
r0
(23) 2
3
pzk> D p 4hk C .1
z0 1 b/
X
z0r
k
hr 5 ;
k > 0:
rkC1
According to (13) and (17), for b D 0; z0 pz> ; 1 z0 1
H.0/ D pz0>
hk D
1 1
z0
h pzk>
i > z0 pzkC1 ;
k > 0:
(24)
If m < 0; then pC .s/ ! pC ; s!0
zs ! 1;
p .s/s
1
!
1 ; jmj.1 b/
and there exist limits pzk D pzk< analogous to (19): pz0< D
pzk< D
1 bqC 1 ! ; jmj.1 b/ b!0 jmj
1 Œp C C .1 jmj.1 b/ k
pz
1 ; jmj
b/Pf C > kg !
b!0
rD
k < 0;
1 Pf C kg; jmj
(25)
k > 0:
Proof. To find the limits pzk> for m > 0; it is necessary to use relation (14). The existence of limits (23) follows from relations (13) and (15). For m < 0; relations (25) are obtained from (14) as a result of the multiplication by s 1 and the limit transition as s ! 0: On the basis of these lemmas, we correct the relations for the generatrices dr .s; / and the limiting distributions for lr .1/: For b D 0; relations (23) and (25) are simplified and agree with the relations for lower semicontinuous .t / obtained earlier. Theorem 2. If .t / is an almost lower semicontinuous process with cumulant (1), then the generatrices dr .s; / and the distributions lr .s / are given by relations (11) and (12) in terms of the probabilities pk .s/ expressed via zs in (14). The corresponding limiting generatrices and distributions lr .1/ are determined depending on the sign of m W (i) for m D 0
zs ! 1 s!0
0; zys ! 1 C 0 ; one can show that s!0
Ee
lr .1/
D 0;
Pflr .1/ D C1g D 1
for any rI
(26)
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(ii) for m > 0; by using the quantities pzk> expressed via z0 in relations (23), one can show that Ee
l0 .1/
D
1 ; 1 C pz0>
lr .1/
Ee
pzr> ; pz0> .pz0> / 1 C
D1
r ¤ 0; (27)
Pflr .1/ > 0g D and, as r D
k!
pzr> ; pz0>
Pflr .1/ > yg D
pzr> y=pz> 0 ; e pz0>
r ¤ 0;
y > 0;
1; lim Pflr .1/ > 0g D 0
and Pflr .1/ D 0g
r! 1
!
r! 1
1I
(iii) for m < 0; by using the values pzk< expressed in (25) via the distribution of the absolute maximum of C ; one can show that Ee
lk .1/
D1
pzk<
pz0<
.pz0< / 1
C
k ¤ 0;
;
(28) Pflk .1/ > 0g D
pzk<
pzk<
Pflk .1/ > yg D
; pz0<
e pz0<
y=p z0<
k ¤ 0I
;
moreover, according to (25), the ratios pzk< =pz0< are independent of k for k < 0 W pzk< pz0<
D
1 1
b ; bqC
Pflk .1/ D 0g D
1
bpC ! 0 bqC b!0
for
kD
r < 1I
(29)
at the same time, for k > 0;
Pflk .1/ > 0g D
pzk< pz0<
Pflk .1/ > yg D
lim Pflk .1/ > yg D 0
k!C1
D
pkC C .1
pzk<
e pz0<
b/Pf C > kg
1 b/ y jmj.1 1 bq C
.y 0/;
bqC
;
;
(30)
y > 0;
Pflk .1/ D 0g
!
k!C1
1:
Proof. Relations (26)–(29) follow from (11) and (12) as a result of the limit transition with regard for the sign of m and the corresponding limiting values of pzk given by Lemmas 2 and 3. It is worth noting that, for m > 0; the atomic probabilities Pflr .1/ D 0g D 1
pzr pz0
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depend on r ¤ 0: We have Pflr .1/ D 0g ! 1 as r ! pz>k D O.z0k /
1185
1 because, according to (23), rD
as
k!
1:
For m < 0; they depend on r only for r > 0: For r < 0; they are independent of r and, for b D 0; they are equal to zero. As k ! C1; we have Pflk .1/ D 0g ! 1 because pkC ! 0 in relation (25) and Pf C > kg ! 0 as k ! C1: Conclusions As the most important corollaries of Theorems 1 and 2 for m ¤ 0; we can mention the fact that the sojourn times in fixed states k ¤ 0 have an exponential distribution [see relations (12) for s > 0 (prior to passing to the limit) and relations (27) and (28) as s ! 0 ] with nonzero atomic probabilities. The parameters of the exponential distribution lr .s / in relation (12) (prior to passing to the limit) are expressed via the distributions of pk .s/ in terms of zs [see (14)]. According to Lemma 3, the limiting parameters are determined by using pzk> for m > 0 and pzk< for m < 0: The most interesting limiting results for the risk process with m < 0 are connected with its sojourn times lr .1/ in the critical states of the risk (“red”) zone with r > u ( u > 0 is the starting capital of an insurance company). According to (30), the quantity pzk< .k > 0/ for the indicated risk process is completely expressed via the distribution of C W prC D Pf C D rg;
r 0:
Moreover, as k ! C1 [see the last two relations in (30)], the sojourn time lk .1/ almost vanishes: Pflk .1/ yg ! 0;
y 0;
Pflk .1/ D 0g ! 1:
i.e.,
However, according to (29), we observe the phenomenon of stable “visiting” of all states k 0 with probabilities independent of k W Pflk .1/ > yg D
1 1
b e bqC
b/ y jmj.1 1 bq C
y 0;
;
Pflk .1/ D 0g D
bpC : 1 bqC
In the case of geometrically distributed demands .n D 1; c1 D c < 1/; the quantities C have the following distribution (close to geometric): pkC D pC zy0 r .1
cy z0 /;
k > 0;
pC D
1
zy0 1 ; 1 c
zy0 D .qC C cpC /
1
:
For m D 0 and s ! 0; the distribution lk .s / (and, hence, lk .t/ as t ! 1 ) becomes degenerate for any k [see (26)]. However, for m" D E" .1/ D c"
." > 0; c ¤ 0/;
as shown in [6], the limiting distribution “"lk;" .1/” with b D 0 and " ! 0 becomes exponential: lim Pf"lk;" .1/ > yg D e
"!0
jcjy
;
y 0:
(31)
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One can readily check the validity of (31) for Z1 lk;" .1/ D
I f" .t/ D kgdt 0
if E" .1/ D m" D c" < 0: Then, in view of (25) and (29) by using the facts that < pz0;" D .1
."/
bqC /Œjcj".1
b/
1
."/ qC D Pf"C > 0g / ; !1 ;
and
"!0
we conclude that Pflk;" .1/ > y"
1
gD
1 1
b ."/ bqC
e
jcj"y="
! e
"!0
jcjy
(31):
for 0 b < 1 and k < 0: For k > 0; it follows from relation (25) that < < lim pzk;" .pz0;" /
"!0
1
D lim
"!0
1
bPf"C > kg 1
."/
bqC
D 1:
Thus, according to (30), relation (31) remains true for k > 0: REFERENCES 1. D. V. Gusak, “Distributions of the sojourn times of a homogeneous process with independent increments over any level,” Dokl. Akad. Nauk Ukr. SSSR, Ser. A, No. 1, 14–17 (1981). 2. D. V. Gusak, “How often is the sum of independent random variables larger than a given number?,” Ukr. Mat. Zh., 34, No. 3, 289–295 (1982); English translation: Ukr. Math. J., 34, No. 3, 234–239 (1982). 3. D. V. Gusak and B. I. Kaplan, “On the distribution of sojourn times in a fixed state for one scheme of random walks with discretely distributed jumps,” in: Analytic Methods in the Problems of Probability Theory [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, 1984, pp. 27–40. 4. D. V. Gusak and A. M. Rozumenko, “Sojourn times in a fixed state for the processes given by the sums of random numbers of discretely distributed terms,” in: Asymptotic Analysis of Random Evolutions [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev, 1994, pp. 74–93. 5. D. V. Gusak, Limiting Problems for the Processes with Independent Increments in the Theory of Risk [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2007). 6. B. I. Kaplan, “Asymptotic behavior of the sojourn time in a fixed state for semicontinuous random walks on the Markov chain,” in: Asymptotic Behavior of the Sums of Random Variables on Markov Processes and Periodic Markov Chains [in Russian], Preprint 85.22, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1965), pp. 50–59.
