Local precise large and moderate deviations for sums of independent random variables

Abstract

Let {X, Xk: k ≥ 1} be a sequence of independent and identically distributed random variables with a common distribution F. In this paper, the authors establish some results on the local precise large and moderate deviation probabilities for partial sums \({S_n} = \sum\limits_{i = 1}^n {{X_i}} \), in a unified form in which x may be a random variable of an arbitrary type, which state that under some suitable conditions, for some constants T > 0, a and τ > 1/2 and for every fixed γ > 0, the relation $$P\left( {{S_n} - na \in \left( {x,\;x + T]} \right)} \right)\~nF\left( {\left( {x + a,\;x + a + T} \right]} \right)$$ holds uniformly for all x ≥ γnτ as n→∞, that is, $$\mathop {\lim }\limits_{n \to + \infty } \mathop {\sup }\limits_{x \geqslant \gamma {n^\tau }} \left| {\frac{{P\left( {{S_n} - na \in \left( {x,\;x + T} \right]} \right)}}{{nF\left( {\left( {x + a,\;x + a + T} \right]} \right)}} - 1} \right| = 0$$ . The authors also discuss the case where X has an infinite mean.