Abstract
Let X 1, X 2, … be independent random variables with EX k= 0 and σk2:=EXk2<∞(k≥ 1) . Set S k= X 1+ ⋯ + X k and assume that sk2:=ESk2→∞ . We prove that under the Kolmogorov condition |Xn|≤Ln,Ln=o(sn/(loglogsn)1/2) we have 1logsn2∑k=1nσk+12sk2f(Sksk)→12π∫Rf(x)e-x2/2dxa.s. for any almost everywhere continuous function f: R→ R satisfying |f(x)|≤eγx2 , 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 {Sn,n≥1} by a Wiener process.
Originalsprache | englisch |
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Seitenumfang | 19 |
Fachzeitschrift | Journal of Theoretical Probability |
Frühes Online-Datum | 6 Mai 2023 |
DOIs | |
Publikationsstatus | Elektronische Veröffentlichung vor Drucklegung. - 6 Mai 2023 |
ASJC Scopus subject areas
- Mathematik (insg.)
- Statistik und Wahrscheinlichkeit
- Statistik, Wahrscheinlichkeit und Ungewissheit