Local limit theorems and renewal theory with no moments

We study i.i.d. sums T-k of nonnegative variables with index 0 : this means P (tau(1) = n) = phi(n)n(-1), with phi(.) slowly varying, so that E (tau(epsilon)(1)) = infinity for all epsilon > 0. We prove a local limit and local (upward) large deviation theorem, giving the asymptotics of P (tau(k) = n) when n is at least the typical length of tau(k). A recent renewal theorem in [22] is an immediate consequence: P (n epsilon tau) similar to P (tau(1) = n)/P (tau(1) > n)(2) as n ->infinity. If instead we only assume regular variation of P (n epsilon tau) and slow variation of U-n : = Sigma(n)(k-0) P(k is an element of tau), we obtain a similar equivalence but with P (tau(1) = n) replaced by its average over a short interval. We give an application to the local asymptotics of the distribution of the first intersection of two independent renewals. We further derive downward moderate and large deviations estimates, that is, the asymptotics of P (tau(k)<= n) when n is much smaller than the typical length of tau(k).