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On an improved Erdő‐Rényi‐type law for increments of partial sums
Author(s) -
Huse Vera R.,
Steinebach Josef
Publication year - 1985
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314953
Subject(s) - combinatorics , sequence (biology) , type (biology) , mathematics , chemistry , ecology , biochemistry , biology
Let X 1 , X 2 ,… be an independently and identically distributed sequence with ξ X 1 = 0, ξ exp ( tX 1 < ∞ ( t ≧ 0) and partial sums S n = X 1 + … + X n . Consider the maximum increment D 1 ( N, K ) = max 0≤ n ≤ N ‐ K ( S n + K ‐ S n )of the sequence ( S n ) in (0, N ) over a time K = K N , 1 ≦ K ≦ N . Under appropriate conditions on ( K N ) it is shown that in the case K N /log N → 0, but K N /(log N ) 1/2 → ∞, there exists a sequence (α N ) such that K ‐1/2 D 1 ( N, K ) ‐ α N converges to 0 w. p. 1. This result provides a small increment analogue to the improved Erdős‐Rényi‐type laws stated by Csörgő and Steinebach (1981).