Local limit theorems for large deviations

We study properties of stable measures with index $\alpha>2$, $\alpha\notin\mathbb N$. Such measures are signed ones and hence they are not probability measures. We show that in some sense these signed measures are limit measures for sums of independent random variables. In the last section of the paper we prove a theorem about large deviations of sums of independent random variables using the positive part of the limit measure.