The central limit theorem for m-dependent variables

Abstract

In a previous paper (4) central limit theorems were obtained for sequences of m-dependent random variables (r.v.'s) asymptotically stationary to second order, the sufficient conditions being akin to the Lindeberg condition (3). In this paper similar theorems are obtained for sequences of m-dependent r.v.'s with bounded variances and with the property that for large n, where s′n is the standard deviation of the nth partial sum of the sequence. The same basic ideas as in (4) are used, but the proofs have been simplified. The results of this paper are examined in relation to earlier ones of Hoeffding and Robbins(5) and of the author (4). The cases of identically distributed r.v.'s and of vector r.v.'s are mentioned.

Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton-Watson trees

“…Then, by the standard central limit for m-dependent variables [23], [14], applied to the random vectors g(ξ i , . .…”

Mean vector testing for high-dimensional dependent observations

“…Extension of the test statistic to the two sample case is provided in Section (4). A finite sample simulation study is performed to evaluate the performance of the proposed test statistic and the results are presented in Section (5). Theoretical details and proofs of the theorems are given in the Appendix.…”

On degenerate sums of m-dependent variables

“…, U i+ −1 ) an -block factor as in Theorem 1.6. Hence the central limit theorem for m-dependent variables [9], [7] yields asymptotic normality of F (T n ), i.e., (3.1) with joint convergence for several T ∈ T * and convergence of first and second moments; this is the method by Devroye [5]. We can now also show that (3.3) holds.…”

“…If we have strict inequality, σ 2 > 0, then Var(S n ) grows linearly; moreover, the classic central limit theorem for m-dependent variables by Hoeffding and Robbins [9] and Diananda [7], see also Bradley [3,Theorem 10.8], shows that…”