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

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
Copyright
Copyright © Applied Probability Trust 1971 

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References

Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
Camm, G. L. (1963) Random gravitational forces in a star field. Monthly Notices Roy. Astronom. Soc. 126, 283298.CrossRefGoogle Scholar
Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Modern Phys. 15, 189.Google Scholar
Daley, D. J. (1970) The non-existence of stationary infinite Newtonian universes and a multi-dimensional model of shot noise. Nature 227, 935.CrossRefGoogle Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Goldman, J. R. (1967) Stochastic point processes: limit theorems. Ann. Math. Statist. 38, 771779.Google Scholar
Kendall, M. G. (1961) A Course in the Geometry of n Dimensions. Griffin, London.Google Scholar
Matern, B. (1960) Spatial Variation. Meddelanden från Statens Skogsforskningsinstitut, 49, No. 5.Google Scholar