On the Independence in the Limit of Sums Depending on the Same Sequence of Independent Random Variables

Let ξt be a stochastic process with independent increments. Suppose that ξt is integervalued 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 νk is the number of discontinuities of ξt of magnitude k in the time interval I = [a, b], then the random variables νk (k = ±1,±2, . . .) are independent.1 This assertion implies, for example, that a homogeneous composed Poisson process ξt may be considered as a superposition of independent ordinary Poisson processes, i.e. can be represented in the form