Almost sure local limit theorem for the Dickman distribution

• Published in: Periodica mathematica Hungarica Vol. 76; no. 2; pp. 155 - 197
• Main Authors: Giuliano, Rita, Szewczak, Zbigniew S., Weber, Michel J. G.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2018; Springer Nature B.V; Springer Verlag
• Subjects: Mathematics; Mathematics and Statistics; Probabilistic models; Probability; Random walk theory; Theorems; Characteristic function; Correlation inequality; General almost sure limit theorem; Dickman function; Bernoulli distribution; Dickman distribution; Almost sure local limit theorem; Poisson distribution; Cumulants; cumulants; Dickman func-tion; almost sure local limit theorem; Bernoulli distribution 2010 Mathematical Subject Classification: Primary 60F15; correlation inequality; 11N37; characteristic function; Secondaries 62H20
• ISSN: 0031-5303; 1588-2829
• DOI: 10.1007/s10998-017-0193-0

We study the asymptotic behavior, and more precisely the second order properties, of the probabilistic model introduced in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002 ) for describing the Dickman distribution. This model appears as an extremal example in the theory of the local and almost sure local limit theorem. We establish a delicate correlation inequality for this system. We apply it to obtain a fine almost sure local limit theorem. In doing so, we also give a corrected proof of the corresponding local limit theorem stated in Hwang and Tsai (Comb Probab Comput 11(4):353–371, 2002 ).