On uniform integrability of random variables

ARTICLE IN PRESS

Statistics & Probability Letters 74 (2005) 272–280 www.elsevier.com/locate/stapro

On uniform integrability of random variables Dariusz Majereka, Wioletta Nowaka, Wies"aw Zi˛ebab, a Department of Mathematics, Technical University, Lublin, Poland Institute of Mathematics, Maria Curie-Sk!odowska University, Lublin 20-031, pl. Marii Curie-Sk!odowskiej 1, Poland

b

Received 9 December 2003; received in revised form 12 October 2004; accepted 1 April 2005 Available online 24 June 2005

Abstract One of the basic theorems used in proving almost sure convergence is the Fatou Lemma. It is common knowledge that any consideration of conditional theorems is a delicate matter. This paper presents a few examples indicating the differences between expected value and conditional expectation. Certain relations between various types of uniform integrability are also provided. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60F15; Secondary 60G48 Keywords: Integrability; Amarts; Almost sure convergence; Conditional expectation

1. Fatou Lemma Let ðO; A; PÞ be a probability space, fFn ; nX1g an increasing sequence of sub-s-fields contained in A, and fX n ; nX1g a sequence of random variable (r.v.) such that X n is Fn measurable for every n. The definition of conditional expectation E F X with respect to the sub-s-fields F  A for X X0 a.s. (almost surely) can be found in Neveu (1975).

Corresponding author.

E-mail addresses: [email protected] (D. Majerek), [email protected] (W. Nowak), [email protected] (W. Zi˛eba). 0167-7152/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2005.04.046

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If minðE F X þ ; E F X  Þo1 a.s., where X þ ¼ maxðX ; 0Þ and X  ¼ maxðX ; 0Þ, then we can define the conditional expectation as follows: E F X ¼ E F X þ  E F X  . In the case of E F jX jo1 a.s. we write X 2 L1F . A random variable t : O ! N ¼ f1; 2; 3; . . .g is a stopping time iff ½t ¼ n 2 Fn for every integer nX1. The set of all finite (P½to1 ¼ 1) stopping times is denoted by T. We denote by T b the set of all bounded stopping times T b ¼ ft 2 T : P½toM ¼ 1 for some M depending on tg. For t 2 T b we define nðtÞ ¼ maxfk : P½t ¼ k40g. Definition 1.1 (Zi˛eba, 1988a). A sequence fX n ; nX1g is uniformly F-integrable if for every Fmeasurable r.v. 40 a.s. there exists an F-measurable r.v. l40 a.s. such that sup E F jX n jI ½jX n j4l o

a.s.

n

One can prove the following theorem. Theorem 1.2 (Zi˛eba, 1988b). A sequence fX n ; nX1g is uniformly F-integrable iff 1. supn E F jX n jo1 a.s. 2. For every F-measurable r.v. 40 a.s. there exists an F-measurable r.v. d40 a.s. such that E F I A od ) sup E F jX n jI A o

a:s.

n

A sequence fX n ; nX1g of r.v. is L1F to an r.v. X if X n ; X 2 L1F and E F jX n  X j ! 0;

n!1

a.s.

Example 1. Let us assume that ðO; A; PÞ ¼ ðh0; 1i2 ; B  B; m2 Þ, where m2 is the two-dimensional Lebesgue’s measure, F1 ¼ B  h0; 1i and F2 ¼ h0; 1i  B. For a random element 8 < 1 ; ðs; tÞað0; 0Þ; X ðs; tÞ ¼ s2 þ t2 : 0; ðs; tÞ ¼ ð0; 0Þ; we have X 2 L1F1 , X 2 L1F2 but X eL1 . Example 2 (Wei-an Zheng, 1980). On the space given above, we can define 8     > < 2k ; o 2 m ; m þ 1 ; o 2 0; 1 ; 1 2 X n ðo1 ; o2 Þ ¼ 2k 2k 2k > :0 otherwise; where n ¼ 2k þ m, and k ¼ maxfi : 2i png, m ¼ 0; 1; . . . ; 2k  1. Then   8 < 1; o 2 m m þ 1 ; o 2 h0; 1Þ; 1 2 E F1 X n ðo1 ; o2 Þ ¼ 2k 2k : 0 otherwise

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and 1 ¼ lim sup E F1 X n XE F1 lim sup X n ¼ 0. The sequence is uniformly integrable EjX n jI ½jX n j42M  o 21M for every nX1. Moreover, it is uniformly F2 -integrable. For the function   1 1 kþ1 for o1 2 h0; 1Þ; o2 2 ; ; k ¼ 1; 2; . . . lðo1 ; o2 Þ ¼ 2 2kþ1 2k we have sup E F2 jX n jI ½jX n j4l ¼ 0. n

But it is not uniformly F1 -integrable sup E F1 jX n jI ½jX n j4l ¼ 1 n

for every F1 -measurable function l. Example 3. Let ðO; A; PÞ ¼ ðh0; 1i2 ; B  B; m2 Þ, F1 ¼ B  h0; 1i and F2 ¼ h0; 1i  B. For 8     > < 2k ; o 2 m ; m þ 1 ; o 2 m ; m þ 1 ; 1 2 X n ðo1 ; o2 Þ ¼ 2k 2k 2k 2k > :0 otherwise; where n ¼ 2k þ m, and k ¼ maxfi : 2i png , m ¼ 0; 1; . . . ; 2k  1 , we have limn X n ¼ 0 a.s. and 1 ¼ lim sup E Fi X n 4E Fi lim sup X n ¼ 0 n

for i ¼ 1; 2.

n

It is easy to verify that fX n ; nX1g is not uniformly Fi -integrable for i ¼ 1; 2. In its classical formulation the Fatou Lemma is as follows: Theorem 1.3. fX n ; nX1g is a sequence of r.v. and X n X0 for nX1; then 1. E lim inf X n p lim inf EX n and if fX n ; nX1g is uniformly integrable (for every 40 there exists M such that supn EjX n jI ½jX n j4M o) then 2. E lim sup X n X lim sup EX n . It is easy to verify that inequality 1 holds when we replace expectation with conditional expectation E F lim inf X n p lim inf E F X n . Wei-an Zheng (1980) showed that the second inequality is not true for conditional expectation (Example 2). The generalization given by Lipster and Shiryayev (1978) is incorrect. In Zi˛eba (1988a) the following theorem was given.

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Theorem 1.4. If fX n ; nX1g is a uniformly F-integrable sequence of r.v., then E F lim sup X n X lim sup E F X n and E F lim inf X n p lim inf E F X n . Hence, we have the following corollary: Corollary 1.5. If fX n ; nX1g is a uniformly F-integrable sequence of r.v. and limn X n ¼ X a.s., then lim E F X n ¼ E F lim X n n

n

a.s.

In view of E F jX j ¼ E F jlimn X n j ¼ E F limn jX n j ¼ E F lim inf n jX n jplim inf n E F jX n jp supn E F jX n jo1 the same theorem implies that X ¼ limn X n is F-integrable. If jX n joY o1 and Y is a F-measurable r.v., then fX n ; nX1g is a uniformly F-integrable sequence. It follows that uniform F-integrability of fX n ; nX1g is necessary for the Fatou Lemma to hold true. Theorem 1.6. If fX n ; nX1g is a sequence of r.v. such that 0pX n 2 L1F , n ¼ 1; 2; . . . and lim sup E F X n pE F lim sup X n o1, then fX n ; nX1g is uniformly F-integrable. Proof. It is obvious that sup E F X n o1. n

Let X  ¼ lim sup X n . Put 40 a.s. (F-measurable) and An ¼ fE F X k oE F X  þ , kXng , nX1. Then An  Anþ1 and limn PðAn Þ ¼ 1. The sequence X 1  X  ; X 2  X  ; . . . ; X n  X  is uniformly F-integrable. Thus, for an F-measurable function 40 a.s. there exists a sequence of Fmeasurable functions fdn ; nX1g such that E F I A odn ) E F jX i  X  jI A o

a.s. for i ¼ 1; 2; . . . ; n.

By the assumption X  2 L1F there exists an F-measurable function d0 such that E F I A od0 ) E F jX  jI A o

a.s.

Let d0 ¼ dn on An nAn1 , nX1, A0 ¼ ; and d ¼ minðd0 ; d0 Þ:

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If E F I A od a.s. then E F X n I A pE F jX n  X  jI A þ E F X  I A 1 X p E F jX n  X  jI Ak nAk1 I A þ  p

k¼1 n X

E F jX n  X  jI Ak nAk1 I A þ

k¼1

p þ

1 X

E F jX n  X  jI Ak nAk1 I A þ 

k¼nþ1 1 X

E F jX n  X  jI Ak nAk1 I A þ 

a.s.

k¼nþ1

If E F I A od then by the F-measurability of I Ak nAk1 we have E F I A I Ak nAk1 odI Ak nAk1 odk and E F jX n  X  jI Ak nAk1 I A o a.s. For kXn þ 1 we have 1 1 X X E F jX n  X  jI Ak nAk1 I A ¼ I Ak nAk1 E F jX n  X  jI A oI Acn p a.s. k¼nþ1

k¼nþ1

Then E X n I A p3 and F

sup E F X n I A p3

a.s.

The proof is complete.

&

n

Without the assumption that E F lim sup X n o1 in (1.6), the theorem will not be true. Example 4. Let ðO; A; PÞ ¼ ðh0; 1i2 ; B  B; m2 Þ, F ¼ h0; 1i  B: For the random variable 8   > < 2k ; o 2 m ; m þ 1 ; o 2 h0; 1Þ; 1 2 X n ðo1 ; o2 Þ ¼ 2k 2k > :0 otherwise; we have the lim sup X n ¼ 1 a.s. and   8 < 1 o 2 h0; 1Þ; o 2 m m þ 1 1 2 E F X n ðo1 ; o2 Þ ¼ 2k 2k : 0 otherwise: Moreover, supn E F jX n jI ½X n 4l ¼ 1 for any F-measurable function l40 a.s., so sequence fX n ; nX1g is not uniformly F-integrable. 2. Amarts Definition 2.1. A sequence fX n ; nX1g is T b -uniformly F-integrable if for every F-measurable r.v. 40 a.s. there exists an F-measurable r.v. l0 40 a.s. such that ess sup E F jX t jI ½jX t j4l0  o

a.s.

t2T b

The definition of ess sup can be found in Neveu (1975) p. 121.

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a:s:

The following example shows that even where the case E F1 X n ! X 0 , n ! 1 and a:s: E X n ! X 0 ; n ! 1 where sðF1 [ F2 Þ ¼ A, the limit limn X n does not exist. F2

Example 5. Let ðO; A; PÞ ¼ ðh0; 1i2 ; B  B; m2 Þ, F1 ¼ B  h0; 1i and F2 ¼ h0; 1i  B. Obviously, sðF1 [ F2 Þ ¼ B  B. Every integer can be written in the form n ¼ 4kðnÞ þ sðnÞ2kðnÞ þ l where kðnÞ ¼ maxfkX0; 4k png, sðnÞ ¼ maxfsX0; s2kðnÞ pn  4kðnÞ g and l ¼ n  4kðnÞ  sðnÞ2kðnÞ . We define     8 l lþ1 < 1; o 2 sðnÞ ; sðnÞ þ 1 ; o 2 ; ; 1 2 X n ðo1 ; o2 Þ ¼ 2kðnÞ 2kðnÞ 2kðnÞ 2kðnÞ : 0 otherwise: Then, lim sup X n ¼ 1 a.s. and lim inf X n ¼ 0 a.s., but   8 < 1 ; o 2 sðnÞ ; sðnÞ þ 1 ; o 2 h0; 1Þ; 1 2 E F1 X n ðo1 ; o2 Þ ¼ 2kðnÞ 2kðnÞ 2kðnÞ : 0 otherwise and

8 < 1 ; F2 E X n ðo1 ; o2 Þ ¼ 2kðnÞ : 0

 o1 2 h0; 1Þ; o2 2

 lþ1 ; ; 2kðnÞ 2kðnÞ l

otherwise

and then limn E F1 X n ¼ limn E F2 X n ¼ 0 a.s. It is easy to notice that the sequence X 0n ¼ 2kðnÞ X n is uniformly F-integrable for F ¼ sð;; OÞ (for every 40 there exists M such that supn EX 0n I ½jX 0n j4M p1=2kðnÞ o1=Mo ) but not T b uniformly F-integrable since supt EX 0t I ½jX 0t j4M ¼ 1 for every M. In Example 3 we have lim sup E F1 X n ¼ lim sup E F2 X n ¼ 1 a.s., lim inf E F1 X n ¼ lim inf E F2 X n ¼ 0 and lim X n ¼ 0 a.s. Definition 2.2 (Zi˛eba, 1991). An adapted sequence fX n ; nX1g of r.v. (X n is Fn -measurable for nX1 ) is called a conditional amart with respect to the s-field F  A iff 1. X n 2 L1F and 2. the net LðE F X t ; X Þ; t 2 T b converges to zero for some r.v. X (where L denotes the Levy–Prokhorov metric). If F is a s-field formed by the subsets of O with measure 0 or 1, then we obtain the following definition of amart (Edgar and Suchestone, 1976). Definition 2.3. A collection fX j ; j 2 Jg of random variables is strongly tight if for any 40 there exists a compact set K  R such that ( ) \ P ½X j 2 K 41  . j2J

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It is known that every almost surely convergent sequence fX n ; nX1gof r.v. is strongly tight (Kruk and Zi˛eba, 1994). Strong tightness performs a similar role with respect to almost sure convergence to measure tightness with respect to weak convergence. Theorem 2.4. If fX n ; nX1g is T b -uniformly integrable then it is strongly tight. Proof. S Assume otherwise. Then there exists 40 such that for every M we have Pf 1 n¼1 ½jX n j4Mg4. For a given M we set 8 1 < inffk; jX j4Mg; S ½jX j4M; k n  tM ¼ n¼1 : 0 otherwise: P Then for every integer k we P have Ak ¼ ½tM ¼ k 2 Fk and 1 k¼1 PðAk Þ4. PðA Þo=2. Now we define tM ¼ minðtM ; k0 Þ. Then There exists k0 such that 1 k k¼k0 EjX tM jI ½jX tM j4M X

kX 0 1

EjX tM jI ½jX tM j4M I Ak

k¼1

¼

kX 0 1

EjX k jI ½jX k j4M I Ak X

k¼1

¼M

kX 0 1

kX 0 1

EMI ½jX k j4M I Ak

k¼1

EI ½jX k j4M I Ak ¼ M

k¼1

kX 0 1 k¼1

 Pð½jX k j4M \ Ak ÞXM . 2

Hence supt2b EjX t jI ½jX t j4M ¼ 1. The proof is complete. & Theorem 2.5. Every uniformly F-integrable conditional amart fX n ; nX1g is T b -uniformly Fintegrable. Proof. We know (see Zi˛eba, 1988b) that every uniformly F-integrable conditional amart is almost sure by convergence to some r.v. X, so sup X s o1 a.s. Moreover, for every given F-measurable 40 there exists an F-measurable function l40 such that E F max X k I ½sups X s 4l o 1pkpn

a.s.

(1)

Hence E F X s I ½X s 4l ¼ E F X s I ½X s 4l;s4n þ E F X s I ½X s 4l;spn pE F X s I ½X s 4l;s4n þ E F max X k I ½X s 4l;spn pE

F

1pkpn X s_nþ1 I ½X s_nþ1 4l;s4n þ E F

max X k I ½sup X k 4l

1pkpn

pE F X s_nþ1 I ½X s_nþ1 4l þ  by (1). It is now necessary to prove T b -uniform F-integrability for s4n.

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Assuming that t0 pn1 , then for tXn1 , 40 a.s. and l40 a.s. we define ( t; X t4l ; t¼ t0 ; X tXl : Therefore, E F X t1 ¼ E F X t I ½X t 4l þ E F X t0 I ½X t pl and hence E F X t I ½X t 4l ¼ E F X t1  E F X t0 þ E F X t0 I ½X t 4l

for any tXn1 .

Moreover, we know that E F X t ! E F X a.s., t ! 1 a.s. Let 40 be F-measurable r.v.; then, Atn ¼ fo : tXtn ; jE F X t  E F X jog;

nptn pnðtn Þ.

If tn otnþ1 and tn ! 1 a.s., then Atn  Atnþ1 and lim P½Atn  ¼ 1. From (2) we now have that for t4tn there exists an F-measurable r.v. lðnÞ 40 such that E F X t I ½X t 4lðnÞ  I Atn o3, since for any t4nðtn Þ E F X t I ½X t 4l ¼ E F X t1  E F X tn þ E F X tn I ½X t 4l ; moreover, I Atn ½E F X t1  E F X tn pI Atn ½ðE F X t1  E F X t Þ þ ðE F X tn  E F X Þo2 and for a fixed tn E F X tn I ½X t 4l o

for l4lðnÞ .

If we define ( lðoÞ ¼

lðnÞ ðoÞ;

o 2 Atnþ1 nAtn ¼ Bn ;

0

o 2 A0 ¼ B0 ;

l ðoÞ;

then E F X t I ½X t 4l ¼ E F X t I ½X t 4l I S1

n¼0

¼

1 X

1 X

E F X t I ½X t 4l\Bn 

n¼0

E F X t I ½X t 4lðnþ1Þ  I Bn o

n¼0

The proof is complete.

Bn

¼

1 X n¼0

&

I Bn 3 ¼ 3

a.s.

(2)

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References Edgar, G.A., Suchestone, L., 1976. Amarts: a class of asymptotic martingales. A discrete parameter. J. Multivariate Anal. 6 (2), 193–221. Kruk, Ł., Zi˛eba, W., 1994. On the tightness of randomly indexed sequences of random elements. Bull. Acad. Polon. Sci. 42 (3), 237–241. Lipster, R.S., Shiryayev, A.N., 1978. Statistics of Random Processes I. Springer, Berlin, Heidelberg, New York. Neveu, J., 1975. Discrete-Parameter Martingales. North-Holland Publishing Company, Amsterdam. Wei-an Zheng, 1980. A note on the convergence of sequence of conditional expectations of random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 53, 291–292. Zi˛eba, W., 1988a. A note of the conditional Fatou Lemma. Probab. Th. Rel. Fields 78, 73–74. Zi˛eba, W., 1988b. On the L1F convergence for conditional amarts. J. Multivariate Anal. 26 (1), 104–110. Zi˛eba, W., 1991. On some properties of conditional amarts. Teoria Veroyatn i ee Primen. 36, 616–617.

Further reading Billingsley, P., 1968. Convergence of Probability Measure. Wiley, New York. Kruk, Ł, Zi˛eba, W., 1985. A criterion of almost sure convergence of asymptotical martingales in banach spaces. Yokohama Mathematical Journal 43, 61–72.