## The Annals of Mathematical Statistics

### A Characterization of Certain Sequences of Noming Constants

Stephen R. Kimbleton

#### Abstract

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.

#### Article information

Source
Ann. Math. Statist., Volume 39, Number 2 (1968), 394-397.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177698403

Digital Object Identifier
doi:10.1214/aoms/1177698403

Mathematical Reviews number (MathSciNet)
MR222934

Zentralblatt MATH identifier
0157.46801

JSTOR