Mod-discrete expansions. (English) Zbl 1416.62131
Summary: We consider approximating expansions for the distribution of integer valued random variables, in circumstances in which convergence in law (without normalization) cannot be expected. The setting is one in which the simplest approximation to the \(n\)-th random variable \(X_n\) is by a particular member \(R_n\) of a given family of distributions, whose variance increases with \(n\). The basic assumption is that the ratio of the characteristic function of \(X_n\) to that of \(R_n\) converges to a limit in a prescribed fashion. Our results cover and extend a number of classical examples in probability, combinatorics and number theory.
MSC:
| 62E17 | Approximations to statistical distributions (nonasymptotic) |
| 60F05 | Central limit and other weak theorems |
| 11N60 | Distribution functions associated with additive and positive multiplicative functions |
| 60C05 | Combinatorial probability |
| 60E10 | Characteristic functions; other transforms |
Keywords:
mod-Poisson convergence; characteristic functions; Poisson-Charlier expansions; Erdős-Kac theoremReferences:
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