Convergence and Precise Asymptotics for Series Involving Self-normalized Sums

Aurel Spătaru ()
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Aurel Spătaru: Romanian Academy

Journal of Theoretical Probability, 2016, vol. 29, issue 1, 267-276

Abstract: Abstract Let $$X, X_{1}, X_{2}, \ldots $$ X , X 1 , X 2 , … be i.i.d. random variables, and set $$S_{n}=X_{1}+\cdots +X_{n}$$ S n = X 1 + ⋯ + X n and $$ V_{n}^{2}=X_{1}^{2}+\cdots +X_{n}^{2}.$$ V n 2 = X 1 2 + ⋯ + X n 2 . Without any moment conditions on $$X$$ X , assuming that $$\{S_{n}/V_{n}\}$$ { S n / V n } is tight, we establish convergence of series of the type (*) $$\sum \nolimits _{n}w_{n}P(\left| S_{n}\right| /V_{n}\ge \varepsilon b_{n}),$$ ∑ n w n P ( S n / V n ≥ ε b n ) , $$\varepsilon >0.$$ ε > 0 . Then, assuming that $$X$$ X is symmetric and belongs to the domain of attraction of a stable law, and choosing $$w_{n}$$ w n and $$b_{n}$$ b n suitably $$,$$ , we derive the precise asymptotic behavior of the series (*) as $$\varepsilon \searrow 0. $$ ε ↘ 0 .

Keywords: Self-normalized sums; Precise asymptotics; Stable law; Domain of attraction; Primary 60G50; Secondary 60F15 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-014-0560-1

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