An extension of renewal theory
1952; American Mathematical Society; Volume: 3; Issue: 2 Linguagem: Inglês
10.1090/s0002-9939-1952-0048734-5
ISSN1088-6826
Autores Tópico(s)Computability, Logic, AI Algorithms
ResumoRenewal theory can be and has been viewed in several different ways. One way is to reduce the problems to those concerning the addition of independent, non-negative random variables having a common distribution. Accordingly we introduce the random variables X1, X2, * * *, independent, non-negative, and all possessing the same distribution function F(x) with mean m-=f.r.xdF(x) (O<m =< X), and their successive sums S. = Et=1 X. Either F(x) is purely discontinuous with all its discontinuities located at the multiples of a fixed real number or it is not so; we call the first case the lattice case and the second the nonlattice case. In the lattice case there is no real loss of generality by assuming (as we shall do in the following) the said discontinuities are all located at integers whose greatest common divisor is one. One of the main results of renewal theory can then be stated as follows: (i) In the lattice case, if x runs through integers:
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