Paper 2011/298

Local limit theorem for large deviations and statistical box-tests

Igor Semaev

Abstract

Let $n$ particles be independently allocated into $N$ boxes, where the $l$-th box appears with the probability $a_l$. Let ${\mu }_r$ be the number of boxes with exactly $r$ particles and $\mu =[{\mu }_{r_1},\dots ,{\mu }_{r_m}]$. Asymptotical behavior of such random variables as $N$ tends to infinity was studied by many authors. It was previously known that if $N{a}_l$ are all upper bounded and $n/N$ is upper and lower bounded by positive constants, then $\mu$ tends in distribution to a multivariate normal low. A stronger statement, namely a large deviation local limit theorem for $\mu$ under the same condition, is here proved. Also all cumulants of $\mu$ are proved to be $O(N)$. Then we study the hypothesis testing that the box distribution is uniform, denoted $$, with a recently introduced box-test. Its statistic is a quadratic form in variables $$. For a wide area of non-uniform $$, an asymptotical relation for the power of the quadratic and linear box-tests, the statistics of the latter are linear functions of $$, is proved. In particular, the quadratic test asymptotically is at least as powerful as any of the linear box-tests, including the well-known empty-box test if $$ is in $$.