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ON THE INDEPENDENCE IN THE LIMIT OF SUMS DEPENDING ON THE SAME SEQUENCE OF INDEPENDENT RANDOM VARIABLES By A. PREKOPA (Budapest) and A. R[~NYI (Budapest), member of the Academy Introduction Let ~ be a stochastic process with independent increments. Suppose lhat ,~ ~ is integer-valued and its sample functions are continuous to the left and have a finite number of discontinuities with probability 1. It can be proved (see [3], Theorem 6) that if ~'j~ is the number of discontinuities of ~ of magnitude k in the time interval 1 ~ [a, b], then the random variables ~,j~ (k ~- :+ 1, --_ 2,...) are independent. ~ This assertion implies, for example, that a homogeneous composed Poisson process ~ may be considered as a superposition of independent ordinary Poisson processes, i. e. can be represented in the form (13 where ~J'~ is an ordinary homogeneous Poisson process, and the processes i are independent (see [4]). For a more general form of this statement see [3]. In w 1 of the present paper we prove a general theorem on the asymptotic independence of certain sums of random variables. w 2 deals with the application of our independence theorem leading to
Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals
Published: Jul 16, 2005
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