Tail asymptotics of the nnth convolution of super-exponential distributions

In this paper we consider distribution density p(x)=e-r(x)(1+o(1)),x→∞ with r(x)⩾0,h(x)=r′(x);h(x)>0,x⩾1. Suppose that pn(x)pn(x) is nn-convolution of p(x)p(x) then in some regularity conditions on r(x)r(x) (in terms of h(x)h(x): slow variation, regular variation and tendency to infinity faster than any power of xx) the following formula is proved: for any fixed n>1n>1 as x→∞x→∞pn(x)=n-1/2(2π)(n-1)/2(r″(x/n))-(n-1)/2exp(-nr(x/n))(1+o(1)).pn(x)=n-1/2(2π)(n-1)/2(r″(x/n))-(n-1)/2exp(-nr(x/n))(1+o(1)).