Estimation of Multivariate Cumulative Processes

Let us suppose that in a result of some action in a random time X we get a random receipt Y. If actions are repeated one-by-one, then in the time interval [0, t] we are interested in the cumulative process of receipts. Let (X, Y) and (Xn, Yn), with n ! 1, be independent equidistributed random vectors. Let X be a positive random variable and Y be a nonnegative integer random variable. We assume that the considered random variables have two moments: E(Xi) " Μi, i " 1, 2, Var(X) " Σ2, E(Y) " Ν1, Var(Y) " Σ &2, and Cov(X, Y) " c. Notations for the distributions are introduced: