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The definition of a multi-dimensional generalization of shot noise

Published online by Cambridge University Press:  14 July 2016

D. J. Daley
D. J. Daley*
Affiliation:
Selwyn College, Cambridge
*
*Now at the Statistics Department, Institute of Advanced Studies, Australian National University, Canberra.
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Abstract

The paper studies the formally defined stochastic process where {tj} is a homogeneous Poisson process in Euclidean n-space En and the a.e. finite Em-valued function f(·) satisfies |f(t)| = g(t) (all |t | = t), g(t) ↓ 0 for all sufficiently large t → ∞, and with either m = 1, or m = n and f(t)/g(t) =t/t. The convergence of the sum at (*) is shown to depend on

  1. (i)

  2. (ii)

  3. (iii)

. Specifically, finiteness of (i) for sufficiently large X implies absolute convergence of (*) almost surely (a.s.); finiteness of (ii) and (iii) implies a.s. convergence of the Cauchy principal value of (*) with the limit of this principal value having a probability distribution independent of t when the limit in (iii) is zero; the finiteness of (ii) alone suffices for the existence of this limiting principal value at t = 0.

Type
Research Papers
Information
Journal of Applied Probability , Volume 8 , Issue 1 , March 1971 , pp. 128 - 135
Copyright
Copyright © Applied Probability Trust 1971 

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