A strong limit theorem expressed by inequalities for the sequences of absolutely continuous random variables

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A strong limit theorem expressed by inequalities for the sequences of absolutely continuous random variables

Wen Liu and Yujin Wang

(Received September 4, 2000) (Revised April 17, 2002)

Abstract. Let fXn;nb1g be an arbitrary sequence of dependent absolutely continuous random viariables,fBn;nb1gbe Borel sets on the real line, andIBnðxÞbe the indicator function of Bn. In this paper, the limit properties of fIBnðXnÞ;nb1g are studied, and a kind of strong limit theorem represented by inequalities with random bounds is obtained.

1. Introduction

Let fXn;nb1g be a sequences of absolutely continuous random variables on the probability spaceðW;F;PÞwith the joint density functiongnðx1;. . .;xnÞ,

n¼1;2;. . .: Let f_kðxkÞ, k¼1;2;. . .; be an arbitrary sequence of density

functions, and call Qⁿ

k¼1

f_kðxkÞ the reference product density. Let

rnðoÞ ¼ Qⁿ

k¼1

f_kðXkÞ

g_nðX1;. . .;X_nÞ if the denominator>0;

0 otherwise,

8>

><

>>

:

ð1Þ

where o is a sample point. In statistical terms, r_nðoÞ is called the likelihood ratio, which is of fundamental importance in the theory of testing the statistical hypotheses (cf. [1, p. 483]; [3, p. 388]). Let

rðoÞ ¼ lim inf

n

1

n lnrnðoÞ ð2Þ

with ln 0¼ y. rðoÞ is called asymptotic log-likelihood ratio. Obviously, r_nðoÞ10 if

2000 Mathematics Subject Classiﬁcation. 60F15, 60F99

Keywords and phrases. Strong limit theorem represented by inequalities, Strong law of large numbers, Likelihood ratio, Supermartingale.

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gnðx1;. . .;xnÞ ¼Yⁿ

k¼1

fkðxkÞ; nb1;

and it will be shown in (13) that rðoÞb0 a.e. in any case. Hence rðoÞ can be used as a random measure of the deviation between the true joint densitygnðx1;. . .;xnÞ ðn¼1;2;. . .Þand the reference product density Qⁿ

k¼1

fkðxkÞ.

Roughly speaking, this deviation may be regarded as the one between fXn;nb1g and the independence case. The smaller rðoÞ is, the smaller the deviation is. The purpose of this paper is to establish a kind of strong limit theorem represented by inequalities with random bounds for the dependent random variables, by using the notion of asymptotic log-likelihood and the martingale convergence theorem, and to extend the analytic technique proposed by Liu [4], [5], and Liu and Yang [6] to the case of absolutely continuous random variables.

2. Main result

Theorem. Let fXn;nb1g, rnðoÞ, rðoÞbe given as above, fBn;nb1g be a sequence of Borel sets of the real line, and IBn be the indicator function of Bn. Let

b¼lim sup

n

1 n

Xⁿ

k¼1

ð

Bk

fkðxkÞdxk ð3Þ

and

D₁ ¼ fo:rðoÞabg; D₂¼ fo:rðoÞbbg:

Then

ðaÞ lim sup

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

a2 ffiffiffiffiffiffiffiffiffiffiffiffi brðoÞ

p þrðoÞ a:e:; ð4Þ

ðbÞ lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

b2 ffiffiffiffiffiffiffiffiffiffiffiffi brðoÞ

p a:e: on D₁; ð5Þ

and

lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

bbrðoÞ a:e: on D₂: ð6Þ

Proof. Let l>0 be a constant, and let

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hkðxkÞ ¼

lfkðxkÞ 1þ ðl1ÞÐ

Bk f_kðxkÞdxk

xkABk; fkðxkÞ

1þ ðl1ÞÐ

Bk fkðxkÞdxk

x_kBB_k: 8>

>>

><

>>

:

ð7Þ

It is easy to see that Qⁿ

k¼1

h_kðxkÞ is a product density function of n variables.

Let

t_nðl;oÞ ¼ 8>

><

>>

: Qⁿ

k¼1

h_kðXkÞ

g_nðX1;. . .;X_nÞ if the denominator>0;

0 otherwise:

ð8Þ

Then tnðl;oÞ is a nonnegative supermartingale that converges a.e. Hence there exists AðlÞAF, PðAðlÞÞ ¼1, such that

lim sup

n

1

n lntnðl;oÞa0; oAAðlÞ: ð9Þ Letting l¼1 in (9), we obtain

lim sup

n

1

n lnrnðoÞa0; oAAð1Þ: ð10Þ This implies that

rðoÞb0; oAAð1Þ: ð11Þ

We have by (7) Yⁿ

k¼1

h_kðXkÞ ¼Yⁿ

k¼1

l^I^Bk^ðx^k^Þf_kðxkÞ 1þ ðl1ÞÐ

Bk fkðxkÞdxk

¼l^T^k¼1ⁿ ^I^Bk^ðx^k^ÞYⁿ

k¼1

f_kðxkÞ 1þ ðl1ÞÐ

Bk fkðxkÞdxk

: ð12Þ

It follows from (1), (8), and (12) that

lnt_nðl;oÞ ¼Xⁿ

k¼1

I_B_kðXkÞlnlXⁿ

k¼1

ln 1þ ðl1Þ ð

Bk

f_kðxkÞdxk

þlnr_nðoÞ:

ð13Þ We have by (9), and (13)

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lim sup

n

1 n

Xⁿ

k¼1

IBkðXkÞlnlXⁿ

k¼1

ln 1þ ðl1Þ ð

Bk

fkðxkÞdxk

þlnrnðoÞ

! a0;

oAAðlÞ: ð14Þ (a) Let l>1. Dividing the two sides of (16) by lnl, we obtain

lim sup

n

1 n

Xⁿ

k¼1

IBkðXkÞ Xⁿ

k¼1

ln½1þ ðl1ÞÐ

Bk f_kðxkÞdxk

lnl þlnrnðoÞ lnl

! a0;

oAAðlÞ: ð15Þ By (15) and (2), we have

lim sup

n

1 n

Xⁿ

k¼1

IBkðXkÞ Xⁿ

k¼1

ln½1þ ðl1ÞÐ

Bk fkðxkÞdxk lnl

! arðoÞ

lnl; oAAðlÞ ð16Þ By (16), (3), the property of the superior limit

lim sup

n ðanb_nÞad)lim sup

n ðanc_nÞalim sup

n ðbnc_nÞ þd; and the inequality 0alnð1þxÞax ðxb0Þ, we have

lim sup

n

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

alim sup

n

1 n

Xⁿ

k¼1

ln½1þ ðl1ÞÐ

Bk f_kðxkÞdxk

lnl

ð

Bk

fkðxkÞdxk

! þrðoÞ

lnl

alim sup

n

1 n

Xⁿ

k¼1

ðl1ÞÐ

Bk f_kðxkÞdxk

lnl

ð

Bk

fkðxkÞdxk

! þrðoÞ

lnl

ab l1 lnl 1

þrðoÞ

lnl; oAAðlÞ: ð17Þ

By using the inequality 1l¹ <lnl ðl>1Þ, we have by (17),

lim sup

n

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

abðl1Þ þlrðoÞ

l1; oAAðlÞ:

ð18Þ

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Let Q be the set of rational numbers in the interval ð1;þyÞ, and let A¼ 7lAQAðlÞ, gðl;rÞ ¼bðl1Þ þlr=ðl1Þ. Then we have by (20),

lim sup

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

agðl;rðoÞÞ; oAA;lAQ: ð19Þ Let b>0. It is easy to see that if r>0, then gðl;rÞ as a function of l attains its smallest value gð1þ ffiffiffiffiffiffiffi

pr=b

;rÞ ¼2 ffiffiffiffiffi pbr

þr on the interval ð1;þyÞ, and gðl;0Þ is increasing on the interval ð1;þyÞ and liml!1þ0gðl;0Þ ¼0.

For each oAAVAð1Þ, if rðoÞ0 y, take l_nðoÞAQ, n¼1;2;. . .; such that lnðoÞ !1þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rðoÞ=b

p . We have by the continuity of g with respect to l,

n!þlimygðlnðoÞ;rðoÞÞ ¼2 ffiffiffiffiffiffiffiffiffiffiffiffi brðoÞ

p þrðoÞ: ð20Þ

By (19),

lim sup

n

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

agðlnðoÞ;rðoÞÞ; n¼1;2;. . .: ð21Þ By (20) and (21),

lim sup

n

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

p þrðoÞ; oAAVAð1Þ:

ð22Þ If rðoÞ ¼y, (22) holds trivially. Since PðAVAð1ÞÞ ¼1, (4) holds by (22) when b>0.

When b¼0, we have by letting l¼e in (19),

lim sup

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

arðoÞ; oAAðeÞ: ð23Þ

Since PðAðeÞÞ ¼1, (4) also holds by (23) when b¼0.

(b) Let 0<l<1. Dividing the two sides of (14) by lnl, we obtain

lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ Xⁿ

k¼1

ln½1þ ðl1ÞÐ

Bk f_kðxkÞdxk

lnl þlnrnðoÞ lnl

! b0;

oAAðlÞ: ð24Þ By (24) and (2), we have

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lim inf

n

1 n

Xⁿ

k¼1

IBkðXkÞ Xⁿ

k¼1

ln½1þ ðl1ÞÐ

Bk fkðxkÞdxk lnl

! brðoÞ

lnl ; oAAðlÞ:

ð25Þ By (25), (3), the property of the inferior limit

lim inf

n ðanb_nÞbd )lim inf

n ðanc_nÞblim inf

n ðbnc_nÞ þd;

and the inequality lnð1þxÞax ð1<xa0Þ, we have

lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

blim inf

n

1 n

Xⁿ

k¼1

ln½1þ ðl1ÞÐ

Bk f_kðxkÞdxk

lnl

ð

Bk

f_kðxkÞdxk

! þrðoÞ

lnl

blim inf

n

1 n

Xⁿ

k¼1

ðl1ÞÐ

Bk fkðxkÞdxk

lnl

ð

Bk

f_kðxkÞdxk

! þrðoÞ

lnl

bb l1 lnl 1

þrðoÞ

lnl; oAAðlÞ: ð26Þ

By using the inequalities 1l¹<lnl<0 and lnl<l1<0 ð0<l<1Þ, we have by (26),

lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

bbðl1Þ þ rðoÞ

l1; oAAðlÞVAð1Þ:

ð27Þ Let Q be the set of rational numbers in the interval ð0;1Þ, and let A¼ 7lAQAðlÞ, hðl;rÞ ¼bðl1Þ þr=ðl1Þ. Then we have by (27),

lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

bhðl;rðoÞÞ; oAAVAð1Þ;lAQ: ð28Þ Let b>0. It is easy to see that if 0<r<b, then hðl;rÞas a function of l attains its largest value hð1 ffiffiffiffiffiffiffi

pr=b

;rÞ ¼ 2 ffiffiffiffiffi pbr

on the interval ð0;1Þ, and hðl;0Þis increasing on the intervalð0;1Þand liml!10hðl;0Þ ¼0, and hðl;bÞ ¼ b½l1þ1=ðl1Þ is decreasing on the interval ð0;1Þ and lim_l!0^þ hðl;bÞ ¼ 2b. For each oAAVAð1ÞVD1, take tnðoÞAQ, n¼1;2;. . .; such that t_nðoÞ !1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rðoÞ=b

p . We have

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n!þlimy hðtnðoÞ;rðoÞÞ ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffi brðoÞ

p : ð29Þ

By (28), we have

lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

bhðtnðoÞ;rðoÞÞ; n¼1;2;. . .: ð30Þ By (29) and (30),

lim inf

n

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

p ; oAAVAð1ÞVD1:

ð31Þ Since PðAVAð1ÞÞ ¼1, (5) holds by (31) when b>0.

When b¼0, rðoÞ ¼0 for oAD₁VAð1Þ. Hence we have by (28),

lim inf

n

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

b0; oAAðlÞVAð1ÞVD1;0<l<1:

ð32Þ Since PðAðlÞVAð1ÞÞ ¼1, (5) also holds by (32) when b¼0.

It is easy to see that when r>bb0, hðl;rÞ as a function of l is decreasing on the interval ð0;1Þ and liml!0^þhðl;rÞ ¼ ðrþbÞ. For each oAAVAð1ÞVD₂, when rðoÞ0 y, take l_nðoÞAQ, n¼1;2;. . .; such that lnðoÞ !0. We have

n!ylim hðlnðoÞ;rðoÞÞ ¼ rðoÞ b: ð33Þ By (28), we have

lim inf

n

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

bhðlnðoÞ;rðoÞÞ; n¼1;2;. . .: ð34Þ It follows from (33) and (34) that,

lim inf

n

1 n

Xⁿ

k¼1

I_B_kðXkÞ ð

Bk

f_kðxkÞdxk

brðoÞ b; oAAVAð1ÞVD₂: ð35Þ Obviously, (35) also holds when rðoÞ ¼y. Since PðAVAð1ÞÞ ¼1, (6) follows from (35) directly.

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3. Some corollaries

Corollary 1. Let B be a Borel set of the real line, SnðB;oÞ be the number of occurrence of X_kð1akanÞ in B, that is,

S_nðB;oÞ ¼Xⁿ

k¼1

I_BðXkÞ:

Then under the conditions of the theorem, we have

lim sup

n

1 n

Xⁿ

k¼1

SnðB;oÞ ð

B

fkðxkÞdxk

p þrðoÞ a:e:;

lim inf

n

1 n

Xⁿ

k¼1

S_nðB;oÞ ð

B

f_kðxkÞdxk

p a:e: on D₁;

lim inf

n

1 n

Xⁿ

k¼1

SnðB;oÞ ð

B

fkðxkÞdxk

bbrðoÞ a:e: on D2:

Proof. Letting Bk ¼B ðk¼1;2;. . .Þ, the corollary follows from the above theorem directly.

The strong law of large numbers for IBnðXnÞ, nb1; is a corollary of the above theorem.

Corollary 2. If fXk;kb1g is independent random variables with density function f_kðxkÞ, then

limn

1 n

Xⁿ

k¼1

IBkðXkÞ ð

Bk

fkðxkÞdxk

¼0 a:e: ð36Þ

Proof. In this case,gnðx1;. . .;xnÞ ¼ Qⁿ

k¼1

fkðxkÞ, andrðoÞ ¼0. Hence (38) follows from (4) and (5) directly.

Acknowledgements

The authors are thankful to the referee for his helpful suggestion.

References

[ 1 ] P. Billingsley, Probability and Measure. Wiley, New York, 1986.

[ 2 ] K. L. Chung, A Course in Probability Theory. Academic Press, New York, 1968.

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[ 3 ] R. G. Laha and V. K. Rohatig, Probability Theory. Wiley, New York, 1979.

[ 4 ] Liu Wen, An analytic technique to prove Borel strong law of large numbers, Amer.

Math. Monthly,98-2(1991), 146–148.

[ 5 ] Liu Wen, Relative entropy densities and a class of limit theorems of the sequences of m-valued random variables, Ann. Probab. 18(1990), 829–839.

[ 6 ] Wen Liu and Weiguo Yang, The Markov approximation of the sequences of N-valued rondom variables and a class of small deviation theorems, Stochastic Process. Appl.

89(2000), 117–130.

Department of Mathematics Hebei University of Technology

Tianjin 300130, CHINA Department of Basic Science Tianjin University of Commerce

Tianjin 300400, CHINA