Almost Sure Limit Theorems for the Pearson Statistic

Almost sure versions of limit theorems by Kruglov for the Pearson χ 2-statistic are obtained. 1. Introduction and preliminary results. Several papers are devoted to almost sure (a.s.) versions of central limit theorems (CLT), for a review see Berkes [1]. In this section we study a new version of the standard procedure (see Lacey and Philipp [5]) to prove an a.s. limit theorem. We use a strong law of large numbers for logarithmically normalized sums and a well-known characterization of the weak convergence but we apply these tools for random elements in a metric space. The setup in Lemma 1 imitates the situation of sums of independent random variables but the scope is much wider. In Sections 2 and 3 we prove a.s. versions of results by Tumanyan [7] and Kruglov [4]. They obtained CLT's for the Pearson χ 2-statistic (used to test goodness of fit) in the case when both the number of observations and the number of classes converge to infinity. We prove a.s. CLT and a.s. functional CLT under the conditions given by Kruglov [4] for the usual CLT. We will denote by