Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-29T16:25:34.229Z Has data issue: false hasContentIssue false
  • English
  • Français

On generalized shot noise

Published online by Cambridge University Press:  01 July 2016

John Rice
John Rice*
Affiliation:
University of California, San Diego
Get access Rights & Permissions [Opens in a new window]

Abstract

A simple expression for the characteristic functional of generalized shot noise is developed. Through expansions in terms of functional derivatives this yields expressions for moment functions of all orders. A central limit theorem also follows. Several examples are discussed.

Keywords

SHOT NOISECHARACTERISTIC FUNCTIONALPOINT PROCESSMOMENT FUNCTIONSSTOCHASTIC PULSES
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 3 , September 1977 , pp. 553 - 565
Copyright
Copyright © Applied Probability Trust 1977 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bartlett, M. S. (1966) An Introduction to Stochastic Processes. Cambridge University Press.Google Scholar
[2] Conference on Physical Aspects of Noise in Electronic Devices. (1968) Peter Peregrinus Ltd., Stevenage, Hertfordshire, England.Google Scholar
[3] Courant, R. and Hilbert, D. (1965) Methods of Mathematical Physics, Vol. 1. Interscience, New York.Google Scholar
[4] Cox, D. R. (1955) Some statistical methods connected with series of events. J. R. Statist. Soc. B 17, 129–164.Google Scholar
[5] Gilchrist, J. and Thomas, J. (1975) A shot process with burst properties. Adv. Appl. Prob. 7, 527–541.Google Scholar
[6] Goldberg, B. and Konovalov, G. (1972) Energy spectra of pulsed random processes of mixed type in systems with statistical multiplexing. Telecommun. Radio Eng. 26–27, 72–77.Google Scholar
[7] Hawkes, A. G. (1971) Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 433–443.Google Scholar
[8] Hawkes, A. G. and Oakes, D. (1974) A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.Google Scholar
[9] Heiden, C. (1969) Power spectrum of stochastic pulse sequences with correlation between the pulse parameters. Phys. Rev. 188, 319–326.Google Scholar
[10] Hopf, E. (1952) Statistical hydromechanics and functional calculus,. J. Rat. Mech. Anal. 1, 87–123.Google Scholar
[11] Karp, S. (1975) Statistical properties of ensembles of classical wave packets. J. Opt. Soc. Amer. 65, 421–424.Google Scholar
[12] Kolmogorov, A. (1935) La transformation de Laplace dans les espaces linéaires. C. R. Acad. Sci. Paris 200, 1717–1718.Google Scholar
[13] Kuno, A. and Ikegaya, K. (1973) A statistical investigation of acoustic power radiated by a flow of random point sources. J. Acoust. Soc. Japan 29, 662–671.Google Scholar
[14] Lauger, P. (1975) Shot noise in ion channels. Biochem. Biophys. Acta 413, 1–10.CrossRefGoogle ScholarPubMed
[15] Lewis, P. A. W. (1964) A branching Poisson process model for the analysis of computer failure patterns. J. R. Statist. Soc. B 26, 398–456.Google Scholar
[16] Lukes, T. (1961) The statistical properties of sequences of stochastic pulses. Proc. Phys. Soc. 78, 153–168.Google Scholar
[17] Milne, R. and Westcott, M. (1972) Further results for Gauss-Poisson processes. Adv. Appl. Prob. 4, 151–176.Google Scholar
[18] Newman, D. S. (1970) A new family of point processes which are characterized by their second moment properties. J. Appl. Prob. 7, 338–358.Google Scholar
[19] Neyman, J. and Scott, E. (1958) A statistical approach to problems of cosmology. J. R. Statist. Soc. B 20, 1–43.Google Scholar
[20] Picinbono, B., Bendjaballah, C. and Pouget, J. (1970) Photoelectron shot noise. J. Math. Phys. 11, 2166–2176.Google Scholar
[21] Rice, S. (1954) Mathematical analysis of random noise. In Selected Papers on Noise and Stochastic Processes, ed. Wax, N., Dover, New York.Google Scholar
[22] Rousseau, M. (1971) Statistical properties of optical glass fields scattered by random media. J. Opt. Soc. Amer. 61, 1307–1316.Google Scholar
[23] Schottky, W. (1918) Über spontane Stromschwankungen in verschieden Elektrizitätsleitern. Ann. Phys. 57, 541–567.Google Scholar
[24] Shiryaev, A. (1960) Some problems in the spectral theory of higher order moments. I. Theory Prob. Appl. 5, 265–284.Google Scholar
[25] Stevens, C. (1972) Inferences about membrane properties from electrical noise measurements. Biophys. J. 12, 1028–1047.Google Scholar
[26] Tapia, R. (1971) The differentiation and integration of nonlinear operators. In Nonlinear Functional Analysis and Applications, ed. Rall, . Academic Press, New York.Google Scholar
[27] Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 1–62.Google Scholar
[28] Verveen, A. and DeFelice, L. (1974) Membrane noise. Prog. Biophys. Mol. Biol. 28, 189–265.Google Scholar
[29] Volterra, V. (1959) Theory of Functionals. Dover, New York.Google Scholar
[30] Westcott, M. (1972) The probability generating functional. J. Austral. Math. Soc. 14, 448–466.CrossRefGoogle Scholar