We give a detailed exposition of the proof Richter's local limit theorem in refined form, and establish stability remainder term this under small perturbations underlying distribution (including smoothing). also discuss related quantitative bounds for characteristic functions Laplace transforms.

We obtain a strong renewal theorem with infinite mean beyond regular variation, when the underlying distribution belongs to domain of geometric partial attraction semistable law index $$\alpha \in (1/2,1]$$ . In process we local limit theorems for both finite and mean, that is, whole range (0,2)$$ also derive asymptotics function (0,1]$$

We prove a local limit theorem for sums of independent random vectors satisfying appropriate tightness assumptions. In particular, the local limit theorem holds in dimension 1 if the summands are uniformly bounded.