Abstract
Let \(\{X_j, j = 1, 2, 3, \ldots\}\) be a sequence of independent, non-degenerate random variables and write
Under quite a diverse variety of conditions we may obtain
as \(n \rightarrow \infty\) for all \(x, - \infty < x < \infty\), and some real \(\mathcal{P} \geqslant 0\). For example, suppose the \(\{X_j\}\) happen to be distributed identically and belong to the domain fo normal attraction of a symmetric stable law with characteristic exponent \(\alpha, 0 < \alpha \leqslant 2, \alpha \neq 1. \, {\rm If} \, EX_j = 0 \, {\rm whenever} E|X_j| < \infty, \) then
as \(n \rightarrow \infty\) for all \(x, - \infty < x < \infty\), as long as \(0 \leqslant \mathcal{P} < 1/\alpha,\) in view of the central limit theory. It is the purpose of this paper to establish some results on the rate of convergence of \({\rm Pr}(S_n < n^p x) \, {\rm to}\, \frac{1}{2}\) by investigating convergence properties of the series
Problems of this type could usefully be called small-deviation problems (following terminology that appears to have been introduced by Borovkov 2).
Received May 12, 1965
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L. E. Baum and M. L. Katz, On the influence of moments on the asymptotic distribution of sums of random variables, Ann. Math. Statist., 34 (1963), 1042–1044.
A. A. Borovkov, Limit theorems on the distribution of maxima of sums of bounded lattice random variables. I, Theor. Probability Appl., 5 (1960), 125–155.
J. Gil-Pelaez, Note on the inversion theorem, Biometrika, 38 (1951), 481–482.
P. L. Hsu, Absolute moments and characteristic functions, J. Chinese Math. Soc, 1 (1951), 259–280.
E. Lukacz, Characteristic functions (London, 1960).
E. J. G. Pitman, Some theorems on characteristic functions of probability distributions, Proc. 4th Berkeley Symp. on Math. Statist, and Prob., 2 (1961), 393–402.
B. Rosén, On the asymptotic distribution of sums of independent identically distributed random variables, Ark. Mat., 4 (1962), 323–332.
F. Spitzer, A Tauberian theorem and its probability interpretation, Trans. Amer. Math. Soc, 94 (1960), 150–169.
— —Principles of random walk (New York, 1964).
J. G. Wendel, The non-absolute convergence of Gil-Pelaez’ inversion integral, Ann. Math. Statist., 32 (1961), 338–339.
D. V. Widder, The Laplace transformation (Princeton, 1964).
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Heyde, C.C. (2010). Some Results on Small-Deviation Probability Convergence Rates for Sums of Independent Random Variables. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_9
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