Timeline for Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator

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Mar 3, 2022 at 0:44	comment	added	Iosif Pinelis		You are right. However, as stated in the answer, "the asymptotics of $p_{S_n}(r)$ will very much depend on how heavy the right tail of the distribution of $X_1$ is". When the tail is exponential-like or lighter than that, then (2.15) is indeed an answer. For tails heavier than exponential and varying regularly enough, one can use results due to A. Nagaev cited in the linked paper.
Mar 2, 2022 at 18:08	vote	accept	Viktor B
Mar 2, 2022 at 18:05	comment	added	Viktor B		Thank you for your answer. I think $\kappa _n$ that I am looking for is contained in (2.15) of the paper that you link. The expression for $\kappa _n$ given in the answer is a little bit unsatisfactory because the question is basically about the density $p_{S_n}$, so the answer should not contain it (otherwise we could take $\kappa _n(x) = p_{S_n}(x)$).
Mar 1, 2022 at 22:35	history	answered	Iosif Pinelis	CC BY-SA 4.0