Precise Asymptotics in Wichura's Law of Iterated Logarithm

  • Divanji Gooty University of Mysore
  • Vidyalaxmi Kokkada University of Mysore
  • K. N. Raviprakash University of Mysore
Keywords: Precise Asymptotics, Rate of Convergence, Law of Iterated Logarithm, Asymmetric StableLaw, Domain of Attraction

Abstract

Let {Xn, 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 Sn=∑nK=1XK. We prove

                                        limε->0(√ε) ∑n≥3(1/n)P(Sn≤(θα-ε)An )=1/(2√2α),

where An = n1/α(log log n)((α-1)/α) θα =(B(α))((α-1)/α) and B(α) = (1 − α)α(α/(1-α)) (cos (πα/2)) (α/α-1)

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Author Biographies

Divanji Gooty, University of Mysore

Department of Studies in Statistics, Manasagangotri, University of Mysore, Mysuru-570006-Karnataka-India

Vidyalaxmi Kokkada, University of Mysore

Department of Studies in Statistics, Manasagangotri, University of Mysore, Mysuru-570006-Karnataka-India

K. N. Raviprakash, University of Mysore

Department of Studies in Statistics, Manasagangotri, University of Mysore, Mysuru-570006-Karnataka-India

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Published
2019-12-21
How to Cite
Gooty, D., Kokkada, V., & Raviprakash, K. N. (2019). Precise Asymptotics in Wichura’s Law of Iterated Logarithm. MathLAB Journal, 4, 172-181. Retrieved from https://purkh.com/index.php/mathlab/article/view/616
Section
Research Articles