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Journal of Zhejiang University SCIENCE A 2001 Vol.2 No.1 P.79-83

http://doi.org/10.1631/jzus.2001.0079


A LIMIT RESULT FOR SELF-NORMALIZED RANDOM SUMS


Author(s):  ZHANG Li-xin, WEN Ji-wei

Affiliation(s):  Department of Mathematics,Zhejiang University,Hangzhou 310028,China

Corresponding email(s): 

Key Words:  self-normalized, i.i.d.random variables, Chernoff function


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ZHANG Li-xin, WEN Ji-wei. A LIMIT RESULT FOR SELF-NORMALIZED RANDOM SUMS[J]. Journal of Zhejiang University Science A, 2001, 2(1): 79-83.

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Abstract: 
Suppose {X,Xn;n≥1} is a sequence i.i.d.r.v. with EX=0 and EX2<∞. Shao (1995) proved a conjecture of Révész (1990): if P(X=±1)=1/2, then ... Furthermore he conjectured that ... In this paper we prove that if supb>0P(X=b)≥P(X=0) then this conjecture is true.

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Reference

[1] Chernoff,H.,1952.A measure of asymptotic efficiency for tests of a hypothesis based on sums of observations,Ann.Math.Statist.,23: 493-507.

[2] Csörgö,M.and Shao,Q.M.,1994.A self-normalized Erdös-Rényi type strong law of large numbers.Stoch.Processes Appl.,50: 187-196.

[3] Erdös,P.and Rényi,A.,1970.On a new law of large numbers. J. Analyze Math., 23: 103-111.

[4] Griffin,P.S.and Kuelbs,J.D.,1989.Self-normalized laws of the iterated logarithm. Ann. Probab., 17: 1571-1601.

[5] Révész,P.,1990.Random Walk in Random and Non-random Environments,World Scientific,Singapore.

[6] Shao,Q.M.,1995.On a conjecture of Révész, Proc. Amer. Soc., 123: 575-582.

[7] Shao,Q.M.,1997.Self-normalized Large Deviations. Ann. Probab., 25: 285-328.

[8] Zhang,L.X.,1998.Sufficient and necessary conditions for limit laws on lag sums of i.i.d.r.v.s.Acta Math.Sinica.New Series. 14: 113-124.

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