A Multidimensional CLT for Maxima of Normed Sums

Abstract

It is shown that if $S_{k,j} = \sum^k_{i = 1} X_{ij}, 1 \leq j \leq d, k \geq 1$ where $(X_{i1}, \cdots, X_{id}), i \geq 1$ are i.i.d. random vectors with positive mean vector $(\mu_1, \cdots, \mu_d)$ and finite covariance matrix $\Sigma$, then for any choice of $\alpha_j$ in $\lbrack 0, 1), 1 \leq j \leq d$ the random vector whose $j$th component is $n^{\alpha_j - 1/2} (\max_{1 \leq k \leq n}S_{k,j}/k^{\alpha_j} - \mu_jn^{1 - \alpha_j})$ converges in law to a multinormal distribution with mean vector zero and covariance matrix $\Sigma$, thereby extending a result of Teicher when $d = 1$.