A remark on the tail probability of a distribution

Robert Chen

Journal of Multivariate Analysis, 1978, vol. 8, issue 2, 328-333

Abstract: Let {Xn}n>=1 be a sequence of independent and identically distributed random variables. For each integer n >= 1 and positive constants r, t, and [epsilon], let Sn = [Sigma]j=1n Xj and E{N[infinity](r, t, [epsilon])} = [Sigma]n=1[infinity] nr-2P{Sn > [epsilon]nr/t}. In this paper, we prove that (1) lim[epsilon]-->0+ [epsilon][alpha](r-1)E{N[infinity](r, t, [epsilon])} =K(r, t) if E(X1) = 0, Var(X1) = 1, and E( X1 t) 0+ G(t, [epsilon])/H(t, [epsilon]) = 0 if 2 0, and E(X1t) [epsilon]n} --> [infinity] as [epsilon] --> 0+ and H(t, [epsilon]) = E{N[infinity](t, t, [epsilon])} = [Sigma]n=1[infinity] nt-2P{ Sn > [epsilon]n2/t} --> [infinity] as [epsilon] --> 0+, i.e., H(t, [epsilon]) goes to infinity much faster than G(t, [epsilon]) as [epsilon] --> 0+ if 2 0, and E( X1 t)

Keywords: The; Erdos-Katz; theorem; the; tail; probability; of; a; distribution; the; standard; normal; random; variable; the; Euler-Maclaurin; sum; formula; the; central; limit; theorem; the; Polya; theorem; the; Toeplitz; lemma (search for similar items in EconPapers)
Date: 1978
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Citations: View citations in EconPapers (13)

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