# On the independence in the limit of sums depending on the same sequence of independent random variables

On the independence in the limit of sums depending on the same sequence of independent random... 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

# On the independence in the limit of sums depending on the same sequence of independent random variables

, Volume 7 (4) – Jul 16, 2005
8 pages

/lp/springer-journals/on-the-independence-in-the-limit-of-sums-depending-on-the-same-ytjZb0BSxh
Publisher
Springer Journals
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF02020528
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Acta Mathematica Academiae Scientiarum HungaricaSpringer Journals

Published: Jul 16, 2005

### References

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