Abstract
Let {X(t), 0<t<∞} be a compound Poisson process so that E{exp (−sX(t))}=exp (−tΦ(s)), where Φ(s)=λ(1−ϕ(s)), λ is the intensity of the Poisson process, and ϕ(s) is the Laplace transform of the distribution of nonnegative jumps. Consider the zero-crossing probability θ=P{X(t)−t=0 for some t,0<t<∞}. We show that θ=Φ′(ω) where ω is the largest nonnegative root of the equation Φ(s)=s. It is conjectured that this result holds more generally for any stochastic process with stationary independent increments and with sample paths that are nondecreasing step functions vanishing at 0.
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References
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, 3rd ed., Vol. I, Wiley, New York.
Nair, V. N., Shepp, L. A. and Klass, M. J. (1986). On the number of crossings of empirical distribution functions, Ann. Probab., 14, 877–890.
Takacs, L. (1967). Combinational Methods in the Theory of Stochastic Processes, Wiley, New York.
Whittaker, E. T. and Watson, G. N. (1927). A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge.
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Mallows, C.L., Nair, V.N. On a zero-crossing probability. Ann Inst Stat Math 41, 1–8 (1989). https://doi.org/10.1007/BF00049104
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DOI: https://doi.org/10.1007/BF00049104