Open Access
April, 1968 A Characterization of Certain Sequences of Noming Constants
Stephen R. Kimbleton
Ann. Math. Statist. 39(2): 394-397 (April, 1968). DOI: 10.1214/aoms/1177698403


Let $\{Y_j\}^\infty_{j = 1}$ be a sequence of random variables defined on a probability space $(\Omega, F, P)$, which are not necessarily independent, or identically distributed. Let $S_n = Y_1 + \cdots + Y_n$. Assume that there exists sequences of constants $\{A_n\}, \{B_n\}, B_n > 0$ such that the limit distribution of $(S_n - A_n)/B_n$ exists. For a class of limit distributions which includes the stable distributions, we give a characterization of $\{B_n\}$ in terms of the dispersion of the sequence of partial sums $S_n$. Such a characterization will be useful in obtaining stable limit theorems for Markov chains since it allows a description of the norming constants which is not dependent on any particular state of the state space of the Markov chain. In addition, using this characterization and that of Tucker in [6], we obtain a partial strengthening of Paul Levy's Theorem on the Augmentation of the Dispersion.


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Stephen R. Kimbleton. "A Characterization of Certain Sequences of Noming Constants." Ann. Math. Statist. 39 (2) 394 - 397, April, 1968.


Published: April, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0157.46801
MathSciNet: MR222934
Digital Object Identifier: 10.1214/aoms/1177698403

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 2 • April, 1968
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