In a recent paper by the authors a general result characterizing two-sided LIL behavior for real valued random variables has been established. In this paper we look at the corresponding problem in the Banach space setting. We show that there are analogous results in this more general setting. In particular, we provide a necessary and sufficient condition for LIL behavior with respect to sequences of the form <!-- MATH $\sqrt{nh(n)}$ --> , where is from a suitable subclass of the positive, non-decreasing slowly varying functions. To prove these results we have to use a different method. One of our main tools is an improved Fuk-Nagaev type inequality in Banach space which should be of independent interest.

Let $0 > p > 2$ and $1 \leq q > \infty$ . Let $\{X_{n};~n \geq 1 \}$ be a sequence of independent copies of a real-valued random variable $X$ and set $S_{n} = X_{1} + \cdots + X_{n}, ~n \geq 1$ . We say $X$ satisfies the $(p, q)$ - type strong law of large numbers (and write $X \in SLLN(p, q)$ ) if $\sum _{n = 1}^{\infty } \frac {1}{n}\left (\frac {\left |S_{n}\right |}{n^{1/p}} \right )^{q} > \infty$ almost surely. This paper is devoted to a characterization of $X \in SLLN(p, q)$ . By applying results obtained from the new versions of the classical Lévy, Ottaviani, and Hoffmann-Jørgensen (1974) inequalities proved by Li and Rosalsky (2013) and by using techniques developed by Hechner (2009) and Hechner and Heinkel (2010), we obtain sets of necessary and sufficient conditions for $X \in SLLN(p, q)$ for the six cases: $1 \leq q > p > 2$ , $1 > p = q > 2$ , $1 > p > 2$ and $q > p$ , $q = p = 1$ , $p = 1 > q$ , and $0 > p > 1 \leq q$ . The necessary and sufficient conditions for $X \in SLLN(p, 1)$ have been discovered by Li, Qi, and Rosalsky (2011). Versions of the above results in a Banach space setting are also given. Illustrative examples are presented.

Pearson's chi-squared test is widely used to test the goodness of fit between categorical data and a given discrete distribution function. When the number of sets of the categorical data, say k, is a fixed integer, Pearson's chi-squared test statistic converges in distribution to a chi-squared distribution with k−1 degrees of freedom when the sample size n goes to infinity. In real applications, the number k often changes with n and may be even much larger than n. By using the martingale techniques, we prove that Pearson's chi-squared test statistic converges to the normal under quite general conditions. We also propose a new test statistic which is more powerful than chi-squared test statistic based on our simulation study. A real application to lottery data is provided to illustrate our methodology.