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ON SECONDARY PROCESSES GENERATED BY A POISSON PROCESS AND THEIR APPLICATIONS IN PHYSICS By LAJOS TAKA.CS (Budapest) (Presented by A. R~NYI) w 1. Formulation of the problem Let us consider a stochastic process of the Poisson type. Let the den- sity of the occurrence of the events be a non-negative, continuous and bounded function /,(u) of the time parameter u. The average number of the events occurring in the time interval (0, t) is, as well known, (1) 3(0 = du and the probability of exactly k events occurring in the time interval (0, t)is (2) P(t, k) e .,e~ [3 (t)] ': (k = 0, 1, 2, .). k! "" The present work is devoted to investigating the following problem. Suppose that every event in the Poisson process is the starting point of a signal (random pulse). Let the progress in time of the amplitude of a signal be f(u,z) where u is the time counted from the beginning of the signal and the parameter Z is a random variable. We suppose that the para- meters belonging to different events are mutually independent rarldom vari- ables with the common distribution function P(Z ~ x)=H(x). We suppose, finally, that
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: Jul 16, 2005
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