## The Annals of Probability

- Ann. Probab.
- Volume 10, Number 2 (1982), 346-373.

### Invariance Principles in Probability for Triangular Arrays of $B$-Valued Random Vectors and Some Applications

#### Abstract

If $\mu_n, \nu$ are probability measures on a separable Banach space, $j_n \rightarrow \infty$ and $\mu^{jn}_n \rightarrow_w \nu$ (so $\nu$ is necessarily infinitely divisible), then it is possible to construct two row-wise independent triangular arrays $\{X_{nj}\}, \{Y_{nj}\}$ such that $\mathscr{L}(X_{nj}) = \mu_n, \mathscr{L}(Y_{nj}) = \nu^{1/jn}$ and $\max_{k \leq jn} \|S_{nk} - T_{nk}\|\rightarrow_\mathrm{P} 0$, where $S_{nk}$ and $T_{nk}$ are the respective partial row sums. Several refinements are proved. These results are applied to establish the weak convergence of the distributions of certain functionals of the partial row sums, improving well-known results of Skorohod. As concrete applications, we prove an arc-sine law for triangular arrays generalizing the Erdos-Kac law and an arc-sine law for strictly stable processes generalizing P. Levy's law for Brownian Motion.

#### Article information

**Source**

Ann. Probab., Volume 10, Number 2 (1982), 346-373.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993862

**Digital Object Identifier**

doi:10.1214/aop/1176993862

**Mathematical Reviews number (MathSciNet)**

MR647509

**Zentralblatt MATH identifier**

0499.60009

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 60J30

**Keywords**

Infinitely divisible measures triangular arrays invariance principle in probability functionals of partial row sums arc-sine laws

#### Citation

de Acosta, Alejandro. Invariance Principles in Probability for Triangular Arrays of $B$-Valued Random Vectors and Some Applications. Ann. Probab. 10 (1982), no. 2, 346--373. doi:10.1214/aop/1176993862. https://projecteuclid.org/euclid.aop/1176993862