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On the Asymptotic Distribution of Large Prime Factors
Author(s) -
Donnelly Peter,
Grimmett Geoffrey
Publication year - 1993
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-47.3.395
Subject(s) - mathematics , simplex , distribution (mathematics) , combinatorics , dirichlet distribution , asymptotic distribution , prime factor , poisson distribution , factorization , integer (computer science) , unit vector , prime (order theory) , discrete mathematics , mathematical analysis , statistics , boundary value problem , algorithm , estimator , computer science , programming language
A random integer N , drawn uniformly from the set (1,2,…, n ), has a prime factorization of the form N = α 1 α 2 …α M where α 1 ⩾ α 2 ⩾ … ⩾ α M . We establish the asymptotic distribution, as n → ∞, of the vector A( n ) = (log α 1 ,/log N : i : ⩾ 1) in a transparent manner. By randomly re‐ordering the components of A( n ), in a size‐biased manner, we obtain a new vector B( n ) whose asymptotic distribution is the GEM distribution with parameter 1; this is a distribution on the infinite‐dimensional simplex of vectors ( x 1 , x 2 ,…) having non‐negative components with unit sum. Using a standard continuity argument, this entails the weak convergence of A( n ) to the corresponding Poisson–Dirichlet distribution on this simplex; this result was obtained by Billingsley [3].