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May, 1982 Invariance Principles in Probability for Triangular Arrays of $B$-Valued Random Vectors and Some Applications
Alejandro de Acosta
Ann. Probab. 10(2): 346-373 (May, 1982). DOI: 10.1214/aop/1176993862


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.


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Alejandro de Acosta. "Invariance Principles in Probability for Triangular Arrays of $B$-Valued Random Vectors and Some Applications." Ann. Probab. 10 (2) 346 - 373, May, 1982.


Published: May, 1982
First available in Project Euclid: 19 April 2007

zbMATH: 0499.60009
MathSciNet: MR647509
Digital Object Identifier: 10.1214/aop/1176993862

Primary: 60F17
Secondary: 60B12 , 60J30

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

Rights: Copyright © 1982 Institute of Mathematical Statistics


Vol.10 • No. 2 • May, 1982
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