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June, 1976 Fluctuations of Sequences which Converge in Distribution
Holger Rootzen
Ann. Probab. 4(3): 456-463 (June, 1976). DOI: 10.1214/aop/1176996094


A sequence $\{Y_n\}^\infty_{n=1}$ of random variables with values in a metric space is mixing with limiting distribution $G$ if $P(\{Y_n \in A\}\mid B) \rightarrow G(A)$ for all $G$-continuity sets $A$ and all events $B$ that have positive probability. It is shown that if $\{Y_n\}$ is mixing with limiting distribution $G$ and if the support of $G$ is separable, then the range $\{Y_n(\omega); n \geqq 1\}$ is dense in the support of $G$ almost surely. A theorem that, under rather general conditions, establishes mixing for the summation processes based on a martingale is given, and as an application it is shown that, under certain conditions, the range of the periodogram is dense in $R^+$ almost surely.


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Holger Rootzen. "Fluctuations of Sequences which Converge in Distribution." Ann. Probab. 4 (3) 456 - 463, June, 1976.


Published: June, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0338.60019
MathSciNet: MR410865
Digital Object Identifier: 10.1214/aop/1176996094

Primary: 60F05
Secondary: 60G17 , 60G35

Keywords: Convergence in distribution , Fluctuations , Martingales , mixing in the sense of Renyi , periodogram

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 3 • June, 1976
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