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A renewal density theorem in the multi-dimensional case

Published online by Cambridge University Press:  14 July 2016

Charles J. Mode
Charles J. Mode*
Affiliation:
Montana State University
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Summary

In this note a renewal density theorem in the multi-dimensional case is formulated and proved. Let f(x) be the density function of a p-dimensional random variable with positive mean vector μ and positive-definite covariance matrix Σ, let hn(x) be the n-fold convolution of f(x) with itself, and set Then for arbitrary choice of integers k1, …, kp–1 distinct or not in the set (1, 2, …, p), it is shown that under certain conditions as all elements in the vector x = (x1, …, xp) become large. In the above expression μ‵ is interpreted as a row vector and μ a column vector. An application to the theory of a class of age-dependent branching processes is also presented.

Type
Research Papers
Information
Journal of Applied Probability , Volume 4 , Issue 1 , November 1967 , pp. 62 - 76
Copyright
Copyright © Applied Probability Trust 

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References

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