Self-normalized Cramér type Moderate Deviations for the Maximum of Sums
نویسندگان
چکیده
Let X1, X2, . . . be independent random variables with zero means and finite variances, and let Sn = ∑n i=1Xi and V 2 n = ∑n i=1X 2 i . A Cramér type moderate deviation for the maximum of the self-normalized sums max1≤k≤n Sk/Vn is obtained. In particular, for identically distributed X1, X2, · · · , it is proved that P(max1≤k≤n Sk ≥ xVn)/(1 − Φ(x)) → 2 uniformly for 0 < x ≤ o(n1/6) under the optimal finite third moment of X1 .
منابع مشابه
Cramér Type Moderate deviations for the Maximum of Self-normalized Sums
Let {X ,X i , i ≥ 1} be i.i.d. random variables, Sk be the partial sum and V 2 n = ∑n i=1 X 2 i . Assume that E(X ) = 0 and E(X )<∞. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that P(max1≤k≤n Sk ≥ x Vn)/(1−Φ(x))→ 2 uniformly in x ∈ [0, o(n)).
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