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Large Deviations for the Partial Sums of Several Random Variable Sequences

Author: WangXueJun
Tutor: HuShuHe
School: Anhui University
Course: Probability Theory and Mathematical Statistics
Keywords: Large deviations ρ-mixing sequence φ-mixing sequence  ~)-mixing sequence  ~)-mixing sequence NA sequence M-Z-type sequence Linear process sequence
CLC: O211
Type: Master's thesis
Year: 2007
Downloads: 74
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Abstract


In recent two centuries, kinds of convergence properties for the partial sums of randomvariable sequences, such as strong law of large numbers and central limit theorem, havebeen the key subjects for probability limit theory research. But large deviations for thepartial sums of random variable sequences have seldom been studied.Let {Xn, n≥1} be a random variable sequences defined on a fixed probability space(Ω,F,μ), and let Sn =sum from i=1 to n Xi, Xn∈Lp, n≥1, 1≤p<∞. If {Xn, n≥1} isindependent and identically distributed(i.i.d.), the weak law of large numbers asserts that (?)μ(|Sn|>nx)=0,x>0.More generally, if the sequence {Xn, n≥1} is stationary(in the strong sense), thenthe ergodic theorem asserts that the result is still true. In recent years, some authors havepaid much attentions to the problem of growth rate ofμ(|Sn|>nx), for example, Na-gaev(Theory Probab.Appl.10(1965), 214-235) got the estimationμ(|Sn|>n)= o(n1-p) forXi∈Lp, 1≤p<∞, and Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159)gave a simple proof that the estimate of Nagaev can’t be improved; If {Xn, n≥1} is a mar-tingale difference sequence, Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159) proved that if supi E(eXi)<∞, then there exists a constant c>0 such thatμ(|Sn|>n)≤e-cn1/3, this bound is optimal for the class of martingale difference sequenceswhich are also strictly stationary and ergodic; If the sequence {Xn, n≥1}is bounded inLp, 2≤p<∞, then Lesigne and Volny (Stochastic Process. Appl. 96(2001), 143-159)got the estimationμ(|Sn|>n)≤cn-p/2 which is again optional for strictly station-ary and ergodic sequences of martingale difference; Yulin Li(Statistics and ProbabilityLetters, 62(2003), 317-321) generalized the result to the case for 1<p≤2, by usingBurkholder’s inequality, Cr-inequality and martingale maximal inequality, he obtainedμ(|Sn|>n)≤cn1-p, these are optimal in a certain sense.In the paper, we study the large deviations for the partial sums ofρ-mixing se- quence,φ-mixing sequence,(?)-mixing sequence,(?)-mixing sequence, NA sequence, M-Z-type sequence and Linear process sequence using some moment inequalities, and obtainthe similar results optimal upper bounds forμ(|Sn|>n) as those for independent andidentically distributed sequence and martingale difference sequence.

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