1972/01/01 by I. Bar-David, A. Nemirovsky, A. M. Nemirovsky · 1 citation
Decision Sciences · #Probabilistic and Robust Engineering Design
paper · doi:10.1109/tit.1972.1054748
An expression for the expectation of level crossings of a class of nondifferentiable shot processes that involve impulse responses having discontinuities is derived. Its first term is essentially the Rician formula [l, pp. 51-53] except that the random variables <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(yo, yo)</tex> in the integrand are conditioned on nonoccurrence of discontinuities and can be interpreted as yielding the contribution of "smooth" crossings; the second term then describes the direct contribution of the jumps. The moments of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(yo, yo)</tex> reflect the influence, at the points of continuity of the process <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y(t)</tex> , of the jumps at its points of discontinuity. Whereas in differentiable processes <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(y, y)</tex> are orthogonal, here <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(yo, yo)</tex> are correlated; furthermore, although <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yo</tex> converges in distribution to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E[yo]</tex> is finite and nonzero in general, evidently <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> has no finite moments. In the Gaussian limit of large densities of counts of the underlying Poisson process, an explicit formula is obtained in terms of known parameters. The Rician term is slightly altered by the correlation between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yo</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yo</tex> . On the other hand, the additional term has a component that increases as the square root of the average number of counts. In various practical cases either the bounded Rician or the diverging term can be dominant. An example of particular interest is examined in detail.