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- Title
ON THE RATES OF THE CHUNG-TYPE LAW OF LOGARITHM.
- Authors
Pang, T.-X.; Lin, Z.-Y.
- Abstract
Let {X, Xn: n ≧ 1} be a sequence of independent identically distributed random variables. Set Sn = X1 + X2 + ⋯ + Xn, Mn = maxk≦n ∣Sk∣, n ≧ 1. By using the strong approximation method we obtain that, if EX = 0, EX2 = σ2 < ∞, and EX2(log∣X∣)b+1 < ∞ for some fixed b > -1, then we have limϵ→ ∞-ϵ-2(b+1) Σ∞n-1 (log n)b/n P(Mn ≦ ϵσ = √(π2n/8 log n) = 4/πΓ (b + 1) Σk=0∞(-1)k/ (2k+1)2b+3. Moreover, the moment convergence, L2 convergence, and a.s. convergence are also discussed.
- Subjects
LOGARITHMIC functions; LAW of large numbers; RANDOM variables; ACCELERATION of convergence in numerical analysis; MULTIVARIATE analysis; LOGARITHMIC integrals
- Publication
Theory of Probability & Its Applications, 2010, Vol 54, Issue 4, p703
- ISSN
0040-585X
- Publication type
Article
- DOI
10.1137/S0040585X97984577