Limit theorems for self-normalized large deviation.

Wang, Qiying

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Let ${X_{n}, n \geq 1}$ be a sequence of independent random variables such that $E X_{n} = a_{n}$ , $E {(X_{n} - a_{n})}^{2} = σ_{n}^{2}$ , $n \geq 1$ . Let ${N_{n}, n \geq 1}$ be a sequence od positive integer-valued random variables. Let us put $S_{N_{n}} = \sum_{k = 1}^{N_{n}} X_{k}$ , $L_{n} = \sum_{k = 1}^{n} a_{k}$ , $s_{n}^{2} = \sum_{k = 1}^{n} σ_{k}^{2}$ , $n \geq 1$ . In this paper we present necessary and sufficient conditions for weak convergence of the sequence ${(S_{N_{n}} - L_{n}) / s_{n}, n \geq 1}$ , as $n \to \infty$ . The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).