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Moment and Probability Bounds with Quasi-Superadditive Structure for the Maximum Partial Sum

F. A. Moricz, R. J. Serfling and W. F. Stout
The Annals of Probability
Vol. 10, No. 4 (Nov., 1982), pp. 1032-1040
Stable URL: http://www.jstor.org/stable/2243558
Page Count: 9
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Moment and Probability Bounds with Quasi-Superadditive Structure for the Maximum Partial Sum
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Abstract

Let X1, ⋯, Xn be arbitrary random variables and put S(i, j) = Xi + ⋯ + Xj and M(i, j) = max {|S(i, i)|, |S(i, i + 1)|, ⋯, |S(i, j)|} for 1 ≤ i ≤ j ≤ n. Bounds for E{exp tM (1, n)}, E Mγ(1, n) and P{M(1, n) ≥ t} are established in terms of assumed bounds for E {exp t|S(i, j)|}, E|S(i, j)|γ and P{|S(i, j)| ≥ t}, respectively. The bounds explicitly involve a nonnegative function g(i, j) assumed to be quasi-superadditive with index Q(1 ≤ Q ≤ 2): g(i, j) + g(j + 1, k) ≤ Q g(i, k), all $1 \leq i \leq j < k \leq n$. Results previously established for the case Q = 1 are improved and are extended to the case $1 < Q < 2$. When g(i, j) is given by $\operatorname{Var} S(i, j)$, applications of the case $Q > 1$ include sequences {Xi} exhibiting long-range dependence, in particular certain self-similar processes such as fractional Brownian motion.

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