and closed under the convergence in probability. If, in addition, the distri­

ANNALES SOCIETATIS MATHEMATICAL POLONAE Series I : COMMENTATIONES MATHEMATICAL X I V (1970) ROCZNIKI POLSKIEGO TO W A R ZYSTW A MATEMATYCZNEGO

Séria I : PRACE MATEMATYCZNE X I Y (1970)

A. K am inski and K . U rba n ik (Wroclaw)

Centered probability distributions

1. Consider a sequence Х г, Х 2, ... of independent random variables with the same probability distribution function F. Throughout this paper we identify random variables which are equal with probability

. Let ^{J iF} be the linear space generated by the random variables X x, X 2, ...

and closed under the convergence in probability. If, in addition, the distri­

bution function F has a finite mean, then S?F will denote the linear space generated by the random variables X 17 X 2, ... and closed under the convergence in mean. It is clear that both spaces JtF and S£F are uniquely determined up to an isomorphism by the distribution function F.

In the sequel by yF we shall denote the characteristic function of

OO

F , i.e. <pF{t) = f eltxF{dx). Further, y F will denote the median of F and Ec

— OO

the distribution function of the constant c.

We say that the distribution function F is centered in probability if and only if

is the only constant random variable belonging to the space JtF . Similarity, we say that F is centered in mean if and only if 0 is the only constant random variable belonging to SPF.

In [3] a prediction problem was investigated for sequences of random variables. In particular, the main theorem states that each stationary completely non-deterministic sequence is a moving average of independ­

ent identically distributed random variables with a centered in probabili­

ty distribution function. Thus the problem of a characterization of centered distribution functions is of interest for the prediction theory. In this note we shall give two characterization of centered distribution functions : one in terms of the probability distributions and the other in terms of their characteristic functions.

We note that for centered in mean distribution functions the solution of our problem is trivial. In fact, by the mean ergodic theorem (see [1]

Chapter

, Section

) the expectations of random variables from ££F belong to ££ f too. Hence it follows that

5 — Prace matematyczne X IV

F is centered in mean if and only if f xF(dx) —

or, equivalently,

~~oo

in terms of the characteristic function,

F is centered in mean if and only if < p ' f {0) = 0.

A characterization of centered in probability distribution functions is less trivial.

*n

2. It is well known that the convergence of akn X k in probability

*=i

to a constant implies that the random variables aknX k (Jc = 1 , . . . , 7cn;

n — 1 , 2 , . . . ) are asymptotically constant (see e.g. [

], p. 114). If, in addition, the random variables X k are not constant, then we have the formula max \akn\ ->

. Thus we have proved the following lemma:

L

2.1. I f the distribution F is not concentrated at a single point kn

and the sequence of random variables £ aknX k tends in probability to a con­

stant, then max \akn\ ->

. k=1

T

2.1. A distribution function F is centered in probability if and only if either F ~ E 0 or

( 2 . 1 ) lim

T— xx)

°° X

T 1 T 4 ^ r 'F ('t o + № )

T

\P f A SjcF(dx-j-y,p)\

>

.

P r o o f . For distribution functions F c of constant random variables the theorem is obvious. Therefore we may assume that F is not concen­

trated at a single point.

Fisst let us suppose that (

.

) holds and F is not centered in prob­

ability. There exist then coefficients akn (7c =

, 2 , . . . , Tcn) n =

,

, . . . ) for which

(

-

)

Ч А ^

* =

holds in probability. Hence, by Lemma

.

, we get the relation (2.3)

(2.4)

(2.5)

max \akn\

Further, (2.2) implies the relations к

2 7

k

—oo

a2 knx 2

1

+ ж

а| F (dx-\- [Ajp) 0,

kn

У, + аы j xF(dx A -»

k = l |a;|< \akn\~

Centered probability distributions 67

(see [

], p. 142). By (

.

) and (2.3) we can find a positive number q and an index n0 such that

DC

I f _к

—1

—oo

e L

l + a2 knx2 F (dx-\- fj,F)

I «*»/**•+«*» / xF(dx-\- ( a f ) fc=i N<l«*nl

for n > n0. Hence and from (2.6) we get the inequality

f

a

— * j - I

F ( dx -f- juF) 0 ,

which contradicts (2.4). The sufficiency of condition (2.1) is thus proved, Now suppose that (

.

) is not satisfied. Then we can choose a sequence Tk tending to oo such that

Tt f X*

lim к— >oo

' ф

F(dx+t*F) -°° -Lk~rX

I Tk ~

/ xF(dx-\- fiF)\

1

-Tk

=

.

Setting Пь

Tk

[ Tk [fiF + J x F (dx+ /г*-)) *] , ^ak = ^{Тк г} sgn% ,

-Tk

where [ж] denotes the integral part of x , we get sequences satisfying the following conditions

lim nk

k— >oo /

^{X +}

F (dx -f- [A f ) =

, ak

lim акпк{ир -{-

Jc— изо j xF(dx-\-fiF))

~ak l

=

.

By a Feller theorem ([2], Chapter 27, Theorem 2) the last statement nk

is equivalent with the convergence ak X,- -> 1 in probability. Thus

?‘=i

1

e which shows that F is not centered in probability. Consequently, condition (2.1) is necessary which completes the proof of the Theorem.

R e m a r k

.

. In the proof of necessity of condition (2.1) the following

statement was established:

I f F is not centered in probability, then there exist a sequence nx, n%, ...

o f positive integers and a sequence ax, a2j ... of real numbers such that nk

in probability.

7 =

It is obvious that symmetric probability distributions are centered in probability. The fact that distributions centered in mean are, in general, not centered in probability is less obvious. Now we shall give a counter­

example.

Put

and

r

du

j --- and G(x) J u2log2u

b r

0{x) = i + j J du u2log2

%F_b(x) for x < e

for x > e .

It is evident that G has a zero mean and, consequently, is centered in mean. Moreover, as a median pa we can take 0. By a simple compu­

tation we get, for T > max(e, b), the formula

T

Г b

xO(dx) = --- — .

J v '

^ogT

Further,

0 0 00

r x 2 Tb2 Tb r dx

T J T2 + x 2 ^ = 2 (T2-\- b2) + ~

~ J jT ^ x ^ l o g ^ x '

— oo e

B y the substitution x = Ту we get the inequality

T

ç dx

dy 1

<

J (T2-\-x2)log2x ~

Tlog2T Moreover,

oo

ç dx

1

dx

J ^(T2

x 2) log2x ^ log2T J T2~\~x2 4Tlog2T

Hence we get the relation T / — 00

In n--- Z7— >oо

X 2

T2 + x 2 G{dx)

T

f xG(dx) I

—T

= 0

Centered probability distributions 69

which, by Theorem 2.1, shows that the distribution function G is not centered in probability.

It is evident that the distribution function G has an infinite va­

riance. For distribution functions with a finite variance we shall prove the following theorem:

T

2 .2 . Distribution functions with a finite variance centered in mean are centered in probability.

oo

P r o o f . Suppose that F has a finite variance and f xF(dx) = 0.

— CO

Since for F = E 0 our assertion is obvious, we may assume that F Ф E 0 and, consequently, the variance of F is positive. Moreover,

and

0

< J x 2F ( d x p F) < о° ,

— O O

T

/ xF{dx-{-pF) J ⁼ J J x F { dx Jr pF) j < — J x 2F(dx-\-yF) ,

\x\ \x\>T

OO J-

f T 4 - x * F ( d X + f t F ) > ~21* / X‘ F ^ + I,r)- T2+ X 2

— OO

Hence we get the relation T f

lim

T 2+ X 2 F (dx-\- /uF) j x 2 F ( d x p F)

_____________ :5î lim — ^_______________

^ UF+ f x F i d x + t,^] x ^ 2 ^ ^ F ( d x + f . F) I _ T

which, by virtue of Theorem 2.1, shows that F is centered in probability.

3. We proceed now to a characterization of centered, in probability distribution functions in terms of their characteristic functions. In what follows SbVg(pF (t) wih denote a continuous branch of the argument of the characteristic function vanishing at the origin.

L

3 .1 . I f

lim f

log\epF(u)\

\'àYg(f>F ( u ) \

du =

,

t->0 t

then the distribution function F is not centered in probability.

P r o o f . Suppose that a sequence alf a2, ... of positive numbers tends to

and

(3 .1 ) lim —

A'—>o o U j c 2

dfc ak /

log\(pF{u)\\

|ar g<pF{u)\

By the continuity of argçoF(t) for every index Jc there exists a number qk such that

< qk <

and

(3.2) bk = [%rg<pF(akqk)\ = max {|arg <^(<

^ )

: 0 < w < 2 } . Setting nk = [fry1], we have the relation

(3.3) lim^fr

.

fc—>oo

Put Tk = ak Xj (k =

,

, .. .) . Denoting by Àk and щ the median

?=i

and the characteristic function of Yk respectively, for each e >

we can choose a positive constant c such that

2

(3.4) I T ( m - * * | » « K c / | l o g | v*(i)||<«

1

(see [1], p. 41). Since

(3.5) 4>M =4>lh(<bt) (* = 1 , 2 , . . . ) , we have, by (3.2), the inequality

2

n 2ak n fr 2a*

{ \^og\y)k {t)\\dt = ~ f |log|^(^)||du< f

J dr, J (Лъ *

%fr/fc f |log|yj(«)l|

l^rgM ^)! du

1 - ak " ak

which, by (3.1), (3.3) and (3.4), imphes the convergence (3.6) lim (Yk— Ik) = 0 in probability.

fc—>00

Hence we get the formula ]im ip k(qk) exp { — ilkqk) =

, because

fc—> oo 0

< qk ^

. Consequently, by (3.5),

lim (nk éhrg(fF (ak qk) - Xk qk) = 0.

fc—>00

Now, taking into account (3.2) and (3.3), we get the formula Km |AÂ ,| qk

fc—>oo

= 1. Since

< <

, we infer that lim |A;,| >

and, consequently, by

fc->oo

(3.6) , lim Ay

Уд.

in probability. But YkeJtF (Jc = 1 , 2 , . . . ) which

fc—>00

implies l e In other words, the distribution function J

is not centered in probability which completes the proof.

L

3.2. I f the distribution function F is not centered in probability, then

2t |loglM*OI|

(3.7) l i m i f

J = » * J |arg<M^)l ^{du =}

^.

Centered probability distributions

71

P r o o f . Suppose that a non-zero constant c belongs to JtF . Then, by Remark 2.1, there exist a sequence of positive integers

and a sequence a17 a2, ... of real numbers tending to

such that nk

ak £ X j -> c in probability. Without loss of generality we may assume

7=1

that all numbers a17 a27 ... are positive, because in the case when a17 a27 ...

contains no subsequence with positive elements we can consider the sequence — a17 —a27 ... and the number —c. Since

lim Fk (akt) = eict

k-> oo

uniformly in every finite interval, we infer that the relations lim % log \<pF(akt)\ =

,

k—>oo

]im.nkBXg(pF(akt) = ct

oo

hold uniformly in every finite interval. Thus

ш f ^{du -}

fc-*» J \^ë< PF(aku)\

which implies (3.7).

As a consequence of Lemmas 3.1 and 3.2 we get the following theorem :

T

3.1. A distribution function F is centered in probability if and only if either F = E 0 or

2

1 , ,

lim —

I t I |arg<pF {u)\ du >

.

Some applications of the above theorem will be given in the next section.

4. Given a distribution function F of a random variable X and a real number c, by F c we shall denote the distribution function of the sum X-j-c, i.e. F c(x) = F ( x — c). Suppose that F has a finite mean value.

It is obvious that m is the mean value of F if and only if F _ m is centered in mean. Furthermore, for all с Ф — m the distribution functions F c are not centered in mean. In an analogous way we can define a substitute of the mean value. Namely, a number r is a substitute of the mean value of F if and only if F _ r is centered in probability and for all с Ф —r the distribution functions F c are not centered in probability.

Now we shall quote two curious examples of distribution functions

which have no substitute of the mean value.

E

. Let F correspond to a symmetric stable distribution of order p (0 < p < 1). The characteristic function of F is given by the formula yF{t) = exp( — a\t\p), where a > 0. For every real number c we have the equation

М

Г г . М = ( И - . ц Г ..

\Wg<PFc(f)\

Hence, by virtue of Theorem 3.1, we infer that for all c the shifted distribution functions F c are centered in probability.

E

2 . Let F be a stable distribution function with the character­

istic function <pF(t) = e xp ( — |

|-HÜog|tf|). For every real number c we have the formula

froglffEcWl

|arg9Fc (<) I c —log i^j| *,

which, by Theorem 3.1, shows that all distribution functions F c are not centered in probability.

The situations described in the above examples and the case when the substitute of the mean value exists are the only possible cases. Namely, we shall prove the following theorem:

T

4 .1 . For each distribution function F one of the following three cases holds :

(i) the substitute of the mean value of F exists,

(ii) for all c the distribution functions Fc are centered in probability, (iii) for all e the distribution functions F c are not centered in probability.

P r o o f . To prove the Theorem it suffices to prove that a centered in probability distribution function F for which F b is not centered in probability for a number b Ф 0 has a substitute of the mean value being equal to 0. In other words, we have to prove that for all с Ф 0 the shifted distribution functions F c are not centered in probability.

From the relation l e JtFb and Bemark

it follows that there exist a sequence пх, п г, ... of integers tending to oo and a sequence ax, a2, ...

of real numbers tending to

such that nk

(4 .1 ) ak (Xj +

) -» 1 in probability.

7 =

Moreover, without loss of generality, we may assume that the limit

(4.2) lim aknk = a

k— >oo

exists, where —о о ф а ф oo.

Centered probability distributions

73

If a — —oo or oo, then, by (4.1),

nk

- (bnk)~l Xj -»

in probability.

7 = 1

Thus 1 e JtF which contradicts the assumption that F is centered in probability. Consequently, — oo < a < oo and, by (4.1) and (4.2),

nk

(4.3) ak J ? Xj -> 1 — ab in probability.

7 = 1

Since F is centered in probability, we infer that the right-hand side of (4.3) is equal to 0, or equivalently,

(4.4) a = b~l .

Given an arbitrary number с Ф 0, we have, by (4.2), (4.3) and (4.4), the relation

nk

ak (Xj + c) -> b~'c in probability.

7 = 1

Thus h~l c e J t Fc and, consequently, the distribution function F e is not centered in probability, which completes the proof.

References

[1] J. L . D o o b , Stochastic processes, New York-London 1953.

[2] B. Y . G n e d e n k o and A . N. K o lm o g o r o v , Limit distributions for sums o f independent random variables, Moscow-Leningrad 1949 (in Russian).

[3] K . U r b a n ik , Prediction of strictly stationary sequences, Coll. Math. 12 (1964)r

pp. 115-129.