On the Complete Moment Convergence for Arrays

鄧總綱

Abstract Let Xni;i≥1,n≥1 be an array of rowwise  mixing random variables. The authors discuss the complete moment convergence for  mixing random variables without assumptions of identical distribution and stochastic domination. The results obtained generalize and improve the corresponding theorems of Hu and Taylor (1997), Zhu (2006), Wu and Zhu (2010).

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Key words arrays of rowwise mixing random variables;complete moment convergence; complete convergence

中圖分類號(hào) AMS(2010) 60F15 文獻(xiàn)標(biāo)識(shí)碼 A

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1 Introduction

The concept of complete convergence was introduced by Hsu and Robbins［1］ as follows: A sequence Xn;n≥1 of random variables is called to converge completely to the constant λ if

∑

SymboleB@ n=1PXn－λ>ε<

SymboleB@  for ε>0. (1)

In view of the BorelCantelli lemma, this implies that Xn→λ almost surely. Therefore the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables. Hsu and Robbins［1］ proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Erd¨os［2］ proved the converse.

The result of HsuRobbinsErd¨os is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. One of the most important generalizations is Baum and Katz［3］ for the strong law of large numbers as follows: Let p≥1α and 12<α≤1. Let Xn;n≥1 be a sequence of i.i.d. random variables with EXn=0. Then the following statements are equivalent:

The desired results (13) and (14) follow from the above statement. This completes the proof of Corollary 1.

On the complete moment convergence for arrays of rowwise mixing random variables in the evaluation of risk estimation、advantage inspection (see Marciniak and Wesolowski (1999) and Fujioka (2011)), reliability (see Gupta and Akman (1998)), life test (see Mendenhall and Lehman (1960)), insurance, financial mathematics (see Ramsay (1993)), complex system (see Jurlewicz and Weron (2002)) and from financial and predict the actual problem and so on all have quite a wide range of applications．

References

［1］ P L HSU, H ROBBINS. Complete convergence and the strong law of large numbers[J]. Proceedings of the National Academy of Sciences of the United States of America, 1947,33:25-31.endprint

[2] P ERDOS. Remark on my paper “on a theorem of Hsu and Robbins”[J]. Annals of Mathematical Statistics, 1950,20:286-291.

[3] L E BAUM, M KATZ. Convergence rates in the law of large numbers[J]. Transactions of American Mathematical Society, 1965,120:108-123.

[4] Y S CHOW . On the rate of moment convergence of sample sums and extremes[J]. Bulletin of the Institute of Mathematics Academia Sinica. 1988.16:177-201.

[5] T C HU, R L TAYLOR. On the strong law for arrays and for the bootstrap mean and variance[J]. International Journal of Mathematics and Mathematical Science, 1997,20(2): 375-382.

[6] M H ZHU. Strong laws of large numbers for arrays of rowwise mixing random variables[J]. Discrete Dynamics in Nature and Society,2007(74296):6.

[7] Y F WU, D J ZHU. Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables[J]. Journal of the Korean Statistical Society, 2000,39（2）:189-197.

[8] S UTEV, M PELIGRAD. Maximal inequalities and an invariance principle for a class of weakly dependent random variables[J]. Journal of Theoretical Probability, 2003,16(1):101-115.endprint

[2] P ERDOS. Remark on my paper “on a theorem of Hsu and Robbins”[J]. Annals of Mathematical Statistics, 1950,20:286-291.

[3] L E BAUM, M KATZ. Convergence rates in the law of large numbers[J]. Transactions of American Mathematical Society, 1965,120:108-123.

[4] Y S CHOW . On the rate of moment convergence of sample sums and extremes[J]. Bulletin of the Institute of Mathematics Academia Sinica. 1988.16:177-201.

[5] T C HU, R L TAYLOR. On the strong law for arrays and for the bootstrap mean and variance[J]. International Journal of Mathematics and Mathematical Science, 1997,20(2): 375-382.

[6] M H ZHU. Strong laws of large numbers for arrays of rowwise mixing random variables[J]. Discrete Dynamics in Nature and Society,2007(74296):6.

[7] Y F WU, D J ZHU. Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables[J]. Journal of the Korean Statistical Society, 2000,39（2）:189-197.

[8] S UTEV, M PELIGRAD. Maximal inequalities and an invariance principle for a class of weakly dependent random variables[J]. Journal of Theoretical Probability, 2003,16(1):101-115.endprint

[2] P ERDOS. Remark on my paper “on a theorem of Hsu and Robbins”[J]. Annals of Mathematical Statistics, 1950,20:286-291.

[3] L E BAUM, M KATZ. Convergence rates in the law of large numbers[J]. Transactions of American Mathematical Society, 1965,120:108-123.

[4] Y S CHOW . On the rate of moment convergence of sample sums and extremes[J]. Bulletin of the Institute of Mathematics Academia Sinica. 1988.16:177-201.

[5] T C HU, R L TAYLOR. On the strong law for arrays and for the bootstrap mean and variance[J]. International Journal of Mathematics and Mathematical Science, 1997,20(2): 375-382.

[6] M H ZHU. Strong laws of large numbers for arrays of rowwise mixing random variables[J]. Discrete Dynamics in Nature and Society,2007(74296):6.

[7] Y F WU, D J ZHU. Convergence properties of partial sums for arrays of rowwise negatively orthant dependent random variables[J]. Journal of the Korean Statistical Society, 2000,39（2）:189-197.

[8] S UTEV, M PELIGRAD. Maximal inequalities and an invariance principle for a class of weakly dependent random variables[J]. Journal of Theoretical Probability, 2003,16(1):101-115.endprint