Journal of Applied Probability

On degenerate sums of m-dependent variables

Svante Janson

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Abstract

It is well known that the central limit theorem holds for partial sums of a stationary sequence (Xi) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if var(Xi) ≠ 0. We show that this happens only in the case when Xi - EXi = Yi - Yi-1 for an (m - 1)-dependent stationary sequence (Yi) with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to be degenerate. Two applications to subtree counts in random trees are given.

Article information

Source
J. Appl. Probab., Volume 52, Number 4 (2015), 1146-1155.

Dates
First available in Project Euclid: 22 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.jap/1450802758

Digital Object Identifier
doi:10.1239/jap/1450802758

Mathematical Reviews number (MathSciNet)
MR3439177

Zentralblatt MATH identifier
1334.60021

Subjects
Primary: 60G10: Stationary processes
Secondary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability

Keywords
m-dependent stationary sequence block factor random tree

Citation

Janson, Svante. On degenerate sums of m -dependent variables. J. Appl. Probab. 52 (2015), no. 4, 1146--1155. doi:10.1239/jap/1450802758. https://projecteuclid.org/euclid.jap/1450802758


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