Tail asymptotics of the nth convolution of super-exponential distributions

In this paper we consider distribution density with . Suppose that pn(x) is n-convolution of p(x) then in some regularity conditions on r(x) (in terms of h(x): slow variation, regular variation and tendency to infinity faster than any power of x) the following formula is proved: for any fixed n>1 as x-->[infinity]pn(x)=n-1/2(2[pi])(n-1)/2(r''(x/n))-(n-1)/2exp(-nr(x/n))(1+o(1)).