Limit Theorem For Random Sums Research Articles

A Limit Theorem for Random Sums of Dependent Indicators and Its Applications in the Theory of Branching Processes

A Limit Theorem for Random Sums of Dependent Indicators and Its Applications in the Theory of Branching Processes

Limit theorems for random sums of dependent d-dimensional random vectors

Let there be given a doubly infinite sequence ~ = { { X , k ; k e N } n e N } of ddimensional random vectors (rv's) defined on a common probability space 4 " (f2,~,,P) and a doubly infinite sequence 5 = { { ,k, ke{0}wN};neN} of o-subfields of ~, which is row-wise increasing ~ a ~1) and adapted to \~ n,k ~ n,k+ 9 , i.e. every X,k is ~,k-measurable. A pair (~, 5) will be called a system. Moreover, let us choose a fixed sequence of random variables vm: (f2, o ~, P)-~ N u { + o o } , meN, such that every v m is a stopping time with respect to the m-th row of ~, i.e. [vmNk]e~m, 1, k e N . Finally, let us define a sequence of random sums: v,

A Weak Convergence Theorem for Random Sums of Independent Random Variables

A limit theorem for random sums of independent random elements of $D\lbrack 0, 1\rbrack$ is proved. The theorem extends and simplifies the proof of a result obtained by R. Pyke in [3]. In a personal communication Professor Pyke remarked to me that the method employed is also the right approach to the problem studied in [4], by adequately defining sequences of random elements of $D\lbrack 0, 1 \rbrack$.