On degenerate sums of m-dependent variables

2013/12/05 by Svante Janson, Janson, Svante
Mathematics · #60C05 #60F05 #60G10 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60C05 #msc:60F05 #msc:60G10

paper · pdf · doi:10.48550/arxiv.1312.1563

11 pages

arxiv created 2013/12/05 · arxiv updated 2013/12/06

Abstract

It is well-known that the central limit theorem holds for partial sums of a stationary sequence (X_i) of m-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if Var(X_i)≠0. We show that this happens only in the case when X_i-\mathbb E X_i=Y_i-Y_i-1 for an (m-1)-dependent stationary sequence (Y_i) 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 degenerate. Two applications to subtree counts in random trees are given.

Related

• The central limit theorem for m-dependent variables
• Central Limit Theorem for m-dependent random variables under sub-linear expectations
• Central limit theorems for associated possibly moving partial sums and application to the non-stationary invariance principle
• Asymptotic Normality of Random Sums of m-dependent Random Variables
• The Gaussian central limit theorem for a stationary time series with infinite variance