TRABAJOS DE ESTADISTICA Y DE INVESTIGAClON OPERATIVA Vol. 35. N~m. 1, 1~4, pp. 104 a 111
A NOTE ON P6LYA'S THEOREM
Dinis Pestana Departamento de Estatfstica e Centro de Estaffstica e Aplir da Universidade de Lisboa
Abstract The class of extended P6LYA functions 0 = {~: 4, is a continuous real valued real function, ~(--t) = ~(0 -- ~(0) ~ [0, I], limt~| = cE E [0, I] and 0(It[) is convex} is a convex set. Its extreme points are identiffed, and using Choquet's theorem it is shown that ~ ~ fl has an integral representation of the form ~(Itl)= So~max{0,1 - I t [ y } dO(y), where G is the distribution function of some random variable Y. As on the other hand max [0, 1 - It] } is the characteristic function of an absolutely continuous random variable X with probability density function f ( x ) = (21r)-l(x/2)-2sin2(x/2), we conclude that ~ is the characteristic function of the absolutely continuous random variable Z = X Y , X and Y independent. Hence any ~ E fl is a characteristic function. This proof sheds an interesting light upon P6LYA'S sufficient condition for a given function to be a characteristic function. Key words: Convex characteristic function P61ya's class Choquet's representation extreme points
(*) Recibido, Enero, 1982
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1. T h o u g h there exist some necessary and sufficient conditions for a given function to be the characteristic function o f some real r a n d o m variable (cf. L ~ c s , 1970), eventually with atoms o f probability at -oo a n d / o r at + ao, these results seem to be, in general, completely inappropriate to decide whether a given function is, or is not, a characteristic function. On the other hand, the sufficient condition for a given real valued real function to be a characteristic function due to P6LYA (1949) - - i f 4 is a continuous real valued real function such that 4 ( - t ) = 4 ( 0 -< 1, l i m t - . . 4 ( t ) = c ~ [0, 1] and 4(It]) is convex, then 4 is a characteristic f u n c t i o n - - is easy to apply. Observe that in the original paper of P6LYA (1949) there was the assumption limt-.oof(t) = 0, but this isn't in fact necessary: in the above statement, if limt-.| ~(t) = c 6 [0, 1], there exists an atom of probability c at the origin (and if 4(0) < 1, there exist atoms of probability (1 - 4(0))/2 at - o o and at +oo). Observe futher that the practical interest of P6LYA'S characteristic functions is rather limited, since they correspond to r a n d o m variables without finite variance (an example of theoretically interesting P6I.YA's characteristic functions: symmetric stable characteristic functions with characteristic exponent ot less than 1). In the present paper we put forward a new p r o o f o f P6LYA'S theorem, by showing that any P6tYA function, i.e. any function sarisfying the assumptions in P6LYA theorem, admits an integral representation of Choquet's type. In order to do so, we begin with some preliminaries on convexity, we identify the extreme points o f the (convex) set o f P6LYA functions and, at the end, we exhibid the integral representation refered to above. The arithmetic properties o f P6LYA'S class o f characteristic functions appear in PEST,~A (1979). 2. We shall say that the real valued real function f is convex on an interval I iff (2.1)
fCAltl + )~2t2) ~< hlf(tO + )~2f(t2)
for every t~, t2 6 I, )~l, ~2 >I 0 such that )~1 + )~2 = 1. If f is a continuous function (as it will be the case in the present paper, (2.1) may be replaced by
(2.2)
f[(tl + t2)/2] ~< [f(tx) + f(t2)]/2
t~, t2 E L 105
It is well k n o w n that a convex function on an interval I has lateral derivatives such t h a t f ' ( t - ) ~ < f ' ( t + ) for every t ~ int (I). For further information on convexity, cf. HARDY, LITTLEWOOD and P6LYA (1959) and ROBERTS and VARaERG (1973). 3. We shall say that [2 is a convex set iff
x, y E f l ~ kx + ( 1 - k ) y e f t ,
(3.1)
XE[0,1].
Let ft be a convex set, a 6 ft. We shall say that a is an extreme point o f [2 iff ft - [ a I is still a convex set. In other words, a 6 f] is an extreme point o f ft iff a=klxl
-b k2x2
with
)~1,)~2__>~, ~xl + k 2 =
1,
X 1 , X 2 6 f t =-----'*'
(3.2) = e i t h e r X ~ 6 [ 0 , I] or x l = x 2 = a . 4. Let [2 be a convex set in a locally convex space E (to insure the existence o f sufficiently m a n y functionals in E* to separate points). If ft is compact, then ex(ft) - - t h e set o f extreme points o f f t - - is necessarily non-empty (cf. PHELPS, 1966), and eventually ex(f0 is itself a compact set. If this is the case, it is then possible to establich the following result: Theorem 4.1 (CHOQUET, 1960). Let [2 be a convex set in a locally convex space E. If c0 is compact and ex([2) also is compact, then any continuous linear functional L on fl is representable by (or is the resultand of, or is the barycenter of) a regular probability measure P whose support is ex(fl), i.e.
(4.1)
L ( f ) = rex(a) L(h) dP(h)
L a continuous linear functional on [2, with P[ft - ex(ft)] = 0. In case the elements o f ft are continuous functions f , observe that the evaluation functional Lx(f) = f(x) is a continuous and linear one, and hence that (4.2)
f ( x ) = LxOr) = ~ex(a)Lx(h)dP(h)= ~ex(n)h(x)dP(h),
f 6f]
i.e. every f E f t admits a Choquet type integral representation.
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5. Let e be the set C = {r ~ is a continuous real valued real function, such that r162162 limt-.**r 1] and ~,(Itl) is convex }. It is easily shown that C is a convex set; on the other hand, the evenness, continuity and convexity o f $ imply that $(Itl) is non-negative and non-increasing and that ~ ' exists almost everywhere, being nonpositive and non-decreasing. Let E be the locally convex space o f all even, continuous, real valued real functions f , such that f " existis almost everywhere, and let us considerer the topology o f u n i f o r m convergence. This topology is induced by the countable family o f semi-norms P n ( f ) = supl Ifl)(Ixl)l, n -1 .< Ixl .< n, i = 0, 1]
and hence E is metrizable; it then follows that any subset o f E is compact iff it is closed and b o u n d e d in E. L e m m a 5.1. e is compact in E. PROOF. It is obvious that e is closed in E. On the other hand, a straightforward application o f the m e a n value t h e o r e m easily shows that sup{ If'(Ixl)l, n - ' .< Ixl -< n, f E e } is finite for n = 1, 2 . . . . . since 2 / a is an upper b o u n d for - f ' ( i x [ ) for x 6 R - ] - a, a[, u > 0, and hence E is b o u n d e d in E. 6. It is quite obvious that C0(t) -- 1 and ~| -- 0 are extreme points o f the convex set C - we shall, from now on, refer to them as the degenerate extreme points o f C. On the other hand, it is straightforward to show that if ~bis a non-degenerate extreme point o f e then ~(0) = 1 and limt-. ~ ~(t) = O. In fact, if that was not so, we would have respectively ~(t) = (1 - ~(O))~| + ~(O)~k(t) and ~(t) = [1 - limt-.| ~(t)l~| + + limt-. | r with ~b = [~(0)] - 1~, 0 = [limt-, ~ ~(t)] - 1~ E C, and this contradicts the hypothesis ~ E ex(e). In what concerns the non-degenerate extreme points o f ~ we m a y further establish the results that follow:
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Lemma 6.1. The real valued real functions (6.1)
$,(t) = m a x l 0 , 1 -
Itl/a},
a>0
are extreme points o f (g. PI~OOF. It is obvious that ~, 6 (2, v a > 0. Let us assume that there exist ~bl, ~b26 ~ , kt, h2 >I 0 (with),1 + X2 = I) such that (6.2)
~ , ( 0 = ~ l f t ( t ) + )*.2f2(t)
Vt E R.
Differentiating we have
(6.3)
~(0 = xlf~(0 + x2~(0
with ~([t[) = -a-tlto,~] and fI(Itl), i = 1, 2 non-positive and non-decreasing. On the other hand (6.4)
f~(Itl)
= )~; l ( _ a - t _)~2f~(ltl))
being the difference between a constant and a non-decreasing function is necessarily non-increasing. But if f~(Itl) is simultaneously nondecreasing and non-increasing, we must conclude that f~(ltl) is constant over [t3, a] - and the same applies to f~(ltl). Hence fit, ~b2 are linear functions over [13,a]. On the other hand, ~bl(0) = f2(0) = I and r = ff2(a) = 0, and a is the least positive real for which f l and f2 take on the value 0, and this implies that f t =" f2 ffi ~,,-i.e. Oa ex(~). (We have discarded the possibility that either f l =" 0 or ~b2---0, since these are trivial cases.) Lemma 6.2. If ~ is a non-degenerate extreme point of e then there exists an a > 0 such that ~(0 ffi ~ , ( 0 = max{0, I - ltl/al. P~ooF. Let ~ be a non-degenerate extreme point of ~ , a = i n f i x R + :O(x)= 01. The fact that ~(0) = 1 and ~([tl) is convex implies that there is a non-negative and non-increasing f u n c t i o n f such that
108
(6.5)
4,(Itl) = 1 - Slotlf(u)du
and, obviously, f(u) = O, vu > a. Let c~ E ]0, a[ be such that
(6.6)
1 -
~_f(u)du > 0 do
and let us define f .
(6.7)
g(u) = I'''(f(u)
u i> ot
and (6.8)
ff(ltl)
= 1 - / f i t ) g ( u ) du. J0
Obviously ~'(Itl)- ff'(Itl) is non-positive and non-decreasing, and hence ~ - ff = ~b* E e . It follows that O = ~k + if* E e and as, by hypothesis, 4)E ex(e), this implies that (6.9)
~b = ) ~ ,
~, E [0, 11
and hence (6.10)
~b' = X~'
a.e.,
The fact that ~'(u) = O'(u) for u >i ~ implies that )~ = 1, a n d hence ~b' = 4)' a.e. O n the other hand, the fact that c~ had been arbitrarily chosen on 10, a[ implies then that ~ ' ( 0 = k (constant) over ]0, a[, and obviously we have that k = - a - 1 . Hence, finally, (6.11)
~(t) = ~,,(t) = max{0, 1 -
Itl/a}.
L e m m a 6.3. The set e x ( e ) is compact. PROOF. F r o m lemmas 6.1 and 6.2 we have that ex((S) = (r
a E [0, oo] }
where
(6.12)
~o(t) -- 1 = m a g i 0 , 1 - [tl/a}, ~| m0
0 < a < oo
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Let us consider the m a p
(6.13)
T: [0, oo] --, C a --* ~ba
Since it is not difficult to show that T is continuous, the fact that ex(C) = T([0, oo]) is the image o f a compact shows that ex(~) is itself compact. 7. In view o f what we have established in the previous paragraphs the hypothesis in C h o q u e t ' s t h e o r e m are satisfied for fl = e . Hence every ~ E e (and in particular every ~b E e such that 4 ( 0 ) = 1) is representable by a regular probability arteasure P supported by ex(C), in the sense that L(~) = Iex(e) L(h) dP(h) for every continuous linear functional L on C. If in particular we consider the evaluation functional Lt(O) = ~(t) - - w h i c h obviously is linear and c o n t i n u o u s - - we conclude that (7. l)
~(t) = Lt(~) = Sex(e) Lt(f) dP(f).
Let us define a measure # over every Borel set B o f [0, ao] as follows: (7.2)
~(B) = P[T(B)]
and put f(a) = # ( [ - ~ o , a ] ) . Now, as Lt(q~,) = max{0, 1 T - l(ex(e)) = [0, oo], we have that (7.3)
ItJ/a} and
4~(t) = ofto,**l max{0, 1 - Jtl/a} dF(a)
or else (7.4)
~(t) = Iomax{O, I - ItJy} dG(y)
where G(y) = 1 - F ( y - i) is the distribution function o f the r a n d o m variable Y = X - i, where X has distribution function F. Observing that ~(0 = max{O, 1 - J t J } is a characteristic function corresponding to a r a n d o m variable W with probability density function fw(x) = ( 2 7 ) - 1(x/2)- 2 sin2(x/2), we conclude that ~ in (7.4) is a characteristic function, corresponding to the absolutely continuous r a n d o m variable Z = WY, W and Y as described above and independent. I10
We have then established, as a n n o u n c e d , the following generalized f o r m o f P6ZYA'S sufficient condition for a given function to be a characteristic function: Theorem 7.1. Let ~b be a continuous real valued real function such that 4~(-t) = ~(t) ~< ~(0) with ~(0) 6 [0, 1], limt-,~,0(t) = c 6 [0, 1], and 4~([tl) convex over ]0, o0[. T h e n r is the characteristic function o f an absolutely continuous r a n d o m variable.
REFERENCES CHOQ~T, G. (1960). Le th~or6me de la representation int6grale dans les ensembles convexes compacts. Ann. Inst. Fourier 10, 333-344. HARDY, G. H., LITTEWOOD,J. E. and PdLYA, G. (1959). Inequah'ties. Cambridge University Press, Cambridge. LuxAcs, E. (1970). Characteristic Functions. Griffin, London. PESTANA, D. (1979). Estrutura quase-d61fica do semigrupo das fun~6es caracterfsticas convexas. Rev. Univ. Santander 2, 999-I001. PrmLPs, R.R. (1966). Lectures on Choquet's Theorem. D.van Nostrand, Princeton. POLYA, G. (1949). Remarks on characteristic functions, in Proceedings o f the First Berkeley Symposium on Mathematical Statistics and Probability, Univ. California Press, Berkeley and Los Angeles. ROnERTS, A. W. and V,Cm3ERG,D. E. (1973). Convex Functions. Academic Press, New York and London.
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