## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 3 (1976), 456-463.

### Fluctuations of Sequences which Converge in Distribution

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 4, Number 3 (1976), 456-463.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996094

**Digital Object Identifier**

doi:10.1214/aop/1176996094

**Mathematical Reviews number (MathSciNet)**

MR410865

**Zentralblatt MATH identifier**

0338.60019

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60G17: Sample path properties 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx]

**Keywords**

Fluctuations convergence in distribution mixing in the sense of Renyi martingales periodogram

#### Citation

Rootzen, Holger. Fluctuations of Sequences which Converge in Distribution. Ann. Probab. 4 (1976), no. 3, 456--463. doi:10.1214/aop/1176996094. https://projecteuclid.org/euclid.aop/1176996094