Abstract
Let (X n ) be a sequence of i.i.d random variables and U n a U-statistic corresponding to a symmetric kernel function h, where h 1(x 1) = Eh(x 1, X 2, X 3, . . . , X m ), μ = E(h(X 1, X 2, . . . , X m )) and ς 1 = Var(h 1(X 1)). Denote \({\gamma=\sqrt{\varsigma_{1}}/\mu}\), the coefficient of variation. Assume that P(h(X 1, X 2, . . . , X m ) > 0) = 1, ς 1 > 0 and E|h(X 1, X 2, . . . , X m )|3 < ∞. We give herein the conditions under which
for a certain family of unbounded measurable functions g, where F(·) is the distribution function of the random variable \({\exp(\sqrt{2} \xi)}\) and ξ is a standard normal random variable.
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References
Berkes I, Csáki E (2001) A universal result in almost sure central limit theory. Stoch Process Appl 94: 105–134
Berkes I, Csáki E, Horváth L (1998) Almost sure limit theorems under minimal conditions. Stat Prob Lett 37: 67–76
Brosamler GA (1988) An almost everywhere central limit theorem. Math Proc Camb Phil Soc 104: 561–574
Feller W (1946) The law of iterated logarithm for identically distributed random variables. Ann Math 47: 631–638
Friedman N, Katz M, Koopmans LH (1966) Convergence rates for the central limit theorem. Proc Natl Acad Sci 56: 1062–1065
Gonchigdanzan K (2005) A note on the almost sure limit theorem for U-statistic. Period Math Hung 50: 149–153
Gonchigdanzan K, Rempala G (2006) A note on the almost sure limit theorem for the product of partial sums. Appl Math Lett 19: 191–196
Holzmann H, Koch S, Min A (2004) Almost sure limit theorems for U-statistics. Stat Prob Lett 69: 261–269
Ibragimov I, Lifshits M (1998) On the convergence of generalized moments in almost sure central limit theorem. Stat Prob Lett 40: 343–351
Li D, Tomkins RJ (1996) Laws of the iterated logarithm for weighted sums of independent random variables. Stat Prob Lett 27: 247–254
Lu C, Qiu J, Xu J (2006) Almost sure central limit theorems for random functions. Sci China 49A: 1788–1799
Petrov V (1975) Sums of independent random variables. Springer, New York
Rempala G, Wesolowski J (2002) Asymptotics for products of sums and U-statistics. Electron Commun Prob 7: 47–54
Schatte P (1988) On strong versions of the central limit theorem. Mathematische Nachrichten 137: 249–256
Serfling W (1980) Approximation theorems of mathematical statistics. Wiley, New York
Wang F, Cheng SH (2003) Almost sure central limit theorem for U-statistics. Chin Ann Math Ser A 24: 735–742
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Peng, Z., Tan, Z. & Nadarajah, S. Almost sure central limit theorem for the products of U-statistics. Metrika 73, 61–76 (2011). https://doi.org/10.1007/s00184-009-0265-0
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DOI: https://doi.org/10.1007/s00184-009-0265-0