Local precise large deviations for sums of random variables with O-regularly varying densities

In this paper we establish a local precise large deviation result for sums Sn, n=1,2,... of independent and identically distributed random variables X1,X2,... with O-regularly varying densities f. The asymptotic behavior of the probability is comparable, for fixed T, with quantities or n(F(x+T)-F(x)).Local precise large deviations O-regularly varying function Regular density Consistently varying tail Dominatedly varying tail