# Precise Asymptotics in Wichura's Law of Iterated Logarithm

## Keywords:

Precise Asymptotics, Rate of Convergence, Law of Iterated Logarithm, Asymmetric StableLaw, Domain of Attraction## Abstract

Let {X_{n}, n ≥ 1} be a sequence of independent and identically distributed random variables with a common distribution function F = P(X ≤ x) in the domain of attraction of an asymmetric stable law, with index α, 1 < α < 2 and set S_{n}=∑^{n}_{K=1}X_{K.} We prove

lim_{ε->0}(√ε) ∑_{n≥3}(1/n)P(Sn≤(θ_{α}-ε)A_{n} )=1/(2√2α),

where A_{n} = n^{1/α}(*log log** *n)^{((α-1)/α)} θ_{α} =(B(α))^{((α-1)/α)} and B(α) = (1 − α)α^{(α/(1-α))} (cos (πα/2)) ^{(α/α-1)}

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## How to Cite

*MathLAB Journal*,

*4*, 172-181. Retrieved from https://purkh.com/index.php/mathlab/article/view/616