## The Annals of Probability

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

Alejandro de Acosta

#### 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

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
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