Abstract
Let {ξi:i≥1} be a sequence of independent, identically distributed (i.i.d. for short) centered random variables. Let Sn=ξ1+⋯+ξn denote the partial sums of {ξi}. We show that sequence {1nmax1≤k≤n|Sk|:n≥1} satisfies the large deviation principle (LDP, for short) with a good rate function under the assumption that P(ξ1≥x) and P(ξ1≤−x) have the same exponential decrease.
Highlights
We show that sequence { max |Sk | : n ≥ 1} satisfies the large deviation principle (LDP, for short) with a good rate n 1≤ k ≤ n function under the assumption that P(ξ 1 ≥ x ) and P(ξ 1 ≤ − x ) have the same exponential decrease
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We shall primarily obtain the result that sequence { max |Sk | : n ≥ 1} of i.i.d. random variables satisfies LDP under n 1≤ k ≤ n the assumption that P(ξ 1 ≥ x ) and P(ξ 1 ≤ − x ) have the same exponential decrease
Summary
Deviations for the Maximum of the Absolute Value of Partial Sums of Random Variables Sequence. Kozlov [8] obtained LDP results by applying a direct probability approach to P( max Sk ≥ nx ) of i.i.d. non-degenerate random variables, which obey 1≤ k ≤ n the Cramer condition. The upper bound estimation for tail probability P( max |Sk | ≥ nx ) for martin1≤ k ≤ n gale differences random variables was obtained by Fan, Grama and Liu [13] in situations where conditional subexponential moments are bounded. N 1≤ k ≤ n only obtained large deviations upper bound To fill this gap, we shall primarily obtain the result that sequence { max |Sk | : n ≥ 1} of i.i.d. random variables satisfies LDP under n 1≤ k ≤ n the assumption that P(ξ 1 ≥ x ) and P(ξ 1 ≤ − x ) have the same exponential decrease
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