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Strong large deviation and local limit theorems. (English) Zbl 0786.60026

Let \(\{Y_ n, n \geq 1\}\) be a sequence of random variables which converge weakly to a random variable \(Y\). The pseudodensity function of \(Y_ n\) is defined by \(q_ n(y;b_ n,S)= {b_ n \over \mu(S)}P(b_ n(Y_ n-y) \in S)\), where \(b_ n \to \infty\), \(\mu\) is the Lebesgue measure on \(R\) and \(S\) is a measurable subset of \(R\) such that \(0<\mu(S)<\infty\). The authors show that if the characteristic function of \(Y_ n\) satisfies some boundedness conditions, then the random variable \(Y\) possesses a bounded and continuous probability density function \(f\), there exists a finite constant \(M\) and an integer \(n=n(S)\) such that \(\sup_ y \{q_ n(y;b_ n,S)\} \leq M\) for \(n\geq n(S)\) and if \(y_ n\to y^*\), then \(q_ n(y;b_ n,S) \to f(y^*)\) as \(n \to \infty\). Analogous results are obtained for lattice valued random variables. These local limit theorems are used to obtain large deviation results for arbitrary and lattice valued random variables under some conditions on their moment generating functions.

MSC:

60F10 Large deviations
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
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