Some Optimal Conditions for the ASCLT.

Autor: Berkes I; A. Rényi Institute of Mathematics, Reáltanoda u. 13-15, Budapest, 1053 Hungary., Hörmann S; Institute of Statistics, Graz University of Technology, Kopernikusgasse 24/III, 8010 Graz, Austria.
Jazyk: angličtina
Zdroj: Journal of theoretical probability [J Theor Probab] 2024; Vol. 37 (1), pp. 209-227. Date of Electronic Publication: 2023 May 06.
DOI: 10.1007/s10959-023-01245-w
Abstrakt: Let X 1 , X 2 , … be independent random variables with E X k = 0 and σ k 2 : = E X k 2 < ∞ ( k ≥ 1 ) . Set S k = X 1 + ⋯ + X k and assume that s k 2 : = E S k 2 → ∞ . We prove that under the Kolmogorov condition | X n | ≤ L n , L n = o ( s n / ( log log s n ) 1 / 2 ) we have 1 log s n 2 ∑ k = 1 n σ k + 1 2 s k 2 f S k s k → 1 2 π ∫ R f ( x ) e - x 2 / 2 d x a . s . for any almost everywhere continuous function f : R → R satisfying | f ( x ) | ≤ e γ x 2 , γ < 1 / 2 . We also show that replacing the o in (1) by O , relation (2) becomes generally false. Finally, in the case when (1) is not assumed, we give an optimal condition for (2) in terms of the remainder term in the Wiener approximation of the partial sum process { S n , n ≥ 1 } by a Wiener process.
(© The Author(s) 2023.)
Databáze: MEDLINE
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