Some exact distributions in traffic noise theory

Marcus, Allan H.

doi:10.2307/1426130

Cited by 18 publications

(7 citation statements)

References 16 publications

(31 reference statements)

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“…Thus, by a transformation of I to L, the pdf of L is It can be shown [2,7] that the distribution of I is approximately lognormal, and the distribution of L approximately normal, over a wide range of parameter values. Thus, by a transformation of I to L, the pdf of L is It can be shown [2,7] that the distribution of I is approximately lognormal, and the distribution of L approximately normal, over a wide range of parameter values.…”

Section: The Inverse Gaussian Distributionmentioning

confidence: 99%

Power Sum Distributions: An Easier Approach Using the Wald Distribution

Marcus¹

1976

Journal of the American Statistical Association

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association.Power sum distributions arise in communications noise and highway noise models. Inverse Gaussian or Wald distributions with the same index of dispersion are similar and infinitely divisible, and often closely approximate lognormal distributions, and, thus, are more suitable than lognormals for theoretical studies. The Wald distribution is an exact model for Poisson distributed point sources with Erlang (0) emissions of noise power.

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Section: The Inverse Gaussian Distributionmentioning

confidence: 99%

Power Sum Distributions: An Easier Approach Using the Wald Distribution

Marcus¹

1976

Journal of the American Statistical Association

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“…Initially, shot noise processes were introduced to model the current induced by a stream of electrons arriving at the anode of a vacuum tube [40]. Since their first appearance in the literature, shot noise processes have been used to model rainfall [36,44], stream-and riverflows [27,45], earthquake occurences [43], computer failures [28], traffic noise [30], delay in claim settlement in insurance [23,24], and several processes in finance [38], to name but a few. The recent paper [1] offers a list of further references.…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for renewal shot noise processes with eventually decreasing response functions

Iksanov

Marynych

Meiners

2014

Stochastic Processes and their Applications

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We consider shot noise processes (X(t)) t≥0 with deterministic response function h and the shots occurring at the renewal epochs 0 = S 0 < S 1 < S 2 . . . of a zero-delayed renewal process. We prove convergence of the finite-dimensional distributions of (X(ut)) u≥0 as t → ∞ in different regimes. If the response function h is directly Riemann integrable, then the finite-dimensional distributions of (X(ut)) u≥0 converge weakly as t → ∞. Neither scaling nor centering are needed in this case. If the response function is eventually decreasing, non-integrable with an integrable power, then, after suitable shifting, the finite-dimensional distributions of the process converge. Again, no scaling is needed. In both cases, the limit is identified. If the distribution of S 1 is in the domain of attraction of an α-stable law and the response function is regularly varying at ∞ with index −β (with 0 ≤ β < 1/α or 0 ≤ β ≤ α, depending on whether E S 1 < ∞ or E S 1 = ∞), then scaling is needed to obtain weak convergence of the finite-dimensional distributions of (X(ut)) u≥0 . The limits are fractionally integrated stable Lévy motions if E S 1 < ∞ and fractionally integrated inverse stable subordinators if E S 1 = ∞.

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“…where t/J is the characteristic function of Z, the relative power emitted by a single vehicle. This has been derived by Marcus (1975); (b) the process X (t) is strictly stationary with covariance function given as…”

Section: Z/(a 2 + X ')mentioning

confidence: 99%