## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 6 (1976), 902-913.

### Sequential Compactness of Certain Sequences of Gaussian Random Variables with Values in $C\lbrack 0, 1 \rbrack$

#### Abstract

Let $(X_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 1)$ be a sequence of Gaussian processes with mean zero and continuous paths on [0, 1] a.s. Let $R_n(t, s) = EX_n(t)X_n(s)$ and suppose that $(R_n: n \geqq 1)$ is uniformly convergent, on the unit square, to a covariance function $R$. It is shown in this paper that under certain conditions the normalized sequence $(Y_n(t): t \in \lbrack 0, 1 \rbrack, n \geqq 2)$ where $Y_n(t) = (2\lg n)^{-\frac{1}{2}}X_n(t)$ is, with probability one, a sequentially compact subset of $C\lbrack 0, 1 \rbrack$ and its set of limit points coincides a.s. with the unit ball in the reproducing kernel Hilbert space generated by $R$. This is Strassen's form of the iterated logarithm in its intrinsic formulation and includes a special case studied by Oodaira in a recent paper.

#### Article information

**Source**

Ann. Probab., Volume 4, Number 6 (1976), 902-913.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995935

**Digital Object Identifier**

doi:10.1214/aop/1176995935

**Mathematical Reviews number (MathSciNet)**

MR428415

**Zentralblatt MATH identifier**

0364.60025

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60G15: Gaussian processes

**Keywords**

Sequential compactness Gaussian processes reproducing kernel Hilbert spaces the law of the iterated logarithm

#### Citation

Mangano, Gian-Carlo. Sequential Compactness of Certain Sequences of Gaussian Random Variables with Values in $C\lbrack 0, 1 \rbrack$. Ann. Probab. 4 (1976), no. 6, 902--913. doi:10.1214/aop/1176995935. https://projecteuclid.org/euclid.aop/1176995935