A Note on Convergence in Probability*

P E D R O F E R N A N D E Z

We give short proofs of general izat ions of the main results conta ined in [1] and [2]. Let S be a metric space with metric d and balls B(x, 6 ) =

= {y : d(y, x) < 5} and 50 the Borel a-field of S. If X is a r andom variable defined on a probabi l i ty space (fl, sJ , P) with values in S we denote with

~ ( X ) the distr ibution of X, and somet imes with 2 ' e ( X ) if more than one probabi l i ty measure is involved. The .symbol ~, (resp. ~ ) indicates weak converge (resp. convergence in probabili ty). If A e ~ is such that P(A) > 0, P( . iA) will indicate the condit ional probabi l i ty given A.

I f P and Q are probabil i t ies on .~' which are mutual ly absolutely cont inuous we will say that they are equivalent and write P - (2.

The symbol A indicates symmetr ic difference; (?C denotes the bounda ry of the set C ~_ S; A c indicates the complemen t of A. For nota t ion and basic propert ies of weak convergence the reader is refered to [3]. The main results are conta ined in Theorems 1 and 2. The proof of T h e o r e m 1 is based on the following Lemma.

LEMMA 1 -- Let (f~, d , ~ ) be a non-atomic f ini te measure space, S a polish

space, v a Borel measure on 5 ~ such that ~(~) = v(S). Then there exists a

mearusable transformation X : f~ -~ S, such that V B ~ 50

~,[X E B] = v(B).

PROOF: It is sufficient to consider the case g(gl) = v(S) = 1. In this case construct a r andom variable Y on ~ such that its distr ibution is uniform on [0, 1]. Then construct a r a n d o m variable with values in S on the pro- babil i ty space ([0, 1], ~ , 2), (where ~ is the Borel a-field of [0, 1] and 2 is the Lebesgue measure) and such that -~f(Z) = v. The composi t ion X = Z Y gives the desired t ransformat ion .

In the next Theo rem S is a polish space.

*Partially supported by the CNPq. *Recebido pela SBM em 2 de maio de 1972.

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THEOREM 1 -- Let (l), ~ , P) be a non-atomic probability space, X a random variable with values in S. Then X , ~ X for all sequences Xn such that Sa(X,) ~ .Sa(X), if and only if X is degenerate.

PROOF: We only prove the "only if" part; the other is well known. If X is not degenerate there exists a Borel set C such that P['X ~ C] > 0 and P[Xq~ C] > 0. Let's assume that P [ X ~ C] >_ 1/2. Let A ~_ [ 'X�9 C] be such that P ( A ) = P(X~ C]. Let X t be a measurable transformation on (A, ~1 r~ A) such that VB �9 6 a

P[X, �9 B] = e ( [ x �9 8] IX r c]).

Let X2 a measurable transformation on (IX r C], ~t c~ IX r C]) such that V B � 9

P[X z ~ B] = P([X �9 B] (n A).

Define - 1' X(o)) if co �9 [X �9 C] - A

Y(co) = { X,(o)) if a) �9 A

/ XE(a~) if o9 �9 IX ~ C]

It is easy to check that &a(y) = s but P l Y # X] > O. If we define X. = Y, Vn, we clearly have s ~ .L,e(X) but X. ~ X.

On atomic spaces this result is false as the following easy example shows.

= {0, 1} P({O} = ~ P({1}) -- ~ X(O) = O, X(1) -- 1.

X is not degenerate and if .~f(X.) ~ .s is easy to verify that X. ~ X at all points.

In the next Theorem S is a separable space.

THEOREM 2 -- X n P-~ X if and only if .~o(X) ~ .~aQ(x), VQ = P.

PRO(.~': The "only if" part follows immediatly. The other implication.is a consequence of the following two lemmas.

LEMMA 2 -- .~e(X,) ~ .L~ao(X), VQ =-- P implies P([X, ~ C] A r x e c ] ) -~ 0 v c such that P[XESC] = O.

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PROOF: Let C be such that P[X �9 OC] = 0. If P[X �9 C] > 0 define

P ( . l [ x � 9 + P Q= 2

Q - P and then by hypothesis Q[X,e Q] ~ Q[x�9 C].

This last equat ion is equivalent to

PEX~�9 C, X�9 C] -* P[X �9 C]. We also have

PEX~e C] ~ PEX �9 C].

If P[X ~ C] = 0 this two equations are trivially satisfyied. It is easyly seen that they imply the result.

L E MMA 3 - - If VC such that P[X �9 OC] = O, P([X~ �9 C] a I X e C]) --+ O, then X~ ~ X.

PROOF : Let e. > 0, 0 < 6 < 1, and A = {(x, y) : (x, y) s S x S and d(x, y) > ~}. Let {x~}i= t. 2 .... be a countable dense subset of S. Select 0 < '2 < e./2 such that P[d(X, x~) = '2, i = 1, 2 . . . . ] = 0. Select now N such that

N

P[X�9 B(x,, y)] > 1-6 i = 1

N

and check that if K = U B(x~, y) i = 1

Therefore

lq

a c~ (K • S)~_ U [B(x,, ~) • S(x,, "2)q. i = l

P[d(X, Xn) > e] < 6 + P[X~ K, d(X,X.) > ~] =~ + P[ (X ,X~) �9 x S)]

N

< (5 + ~ P[(X,X.) �9 y) x B(x,, y)c] i = l

N

= 6 + S P[X �9 S(x,, 7), x ~ ~ S(x, , y)] i = 1

N

<- 6 + 2 P([X ~ B(xi, ~,)] A [Xo s B(x,, "2)])- i =1 .

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Therefore

lim,,sup P[d(X, X,,) > ~:] _< 6 V6 > 0

which is the desired result.

R E F E R E N C E S

[1] A. R. PADMANABHAN - - (1970) -- Some conditions for convergence in probability and convergence in law to coincide. Mathemat ica Japon icae , Vol. 15, n. ~ 2, 105-109.

[2] A. R. PADMANABHAN -- (1970) - Convergence b7 probabili(v and allied results. Mathemat i ca Japonicae , Vol. 15, n. ~ 2, l l l - l l 7 .

[3] B I L L I N G S L E Y , Patrick. ( 1 9 6 8 ) - Convergence O/prohahility me. . ,me , . John Wiley and Sons, New York .

I n s t i t u t o de M a t e m l i t i c a P u r a e A p l i c a d a - - R i o de J a n e i r o

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