Volodya , I Miss You ( Two Correlated Collective Risk Models )

is the total claim amount. In the classical theory it is assumed that (i) N and (Y1, Y2, . . .) are independent random variables (r.v.’s); (ii) Y1, Y2, . . . are independent and (iii) Y1, Y2, . . . have the same distribution. The assumptions (ii) and (iii) of mutual independence between the components of the sum is very convenient, mainly because the mathematics is easier. In many situations, the individual claims (risks) are dependent since they are influenced by the same economic environment. In this study we relax the condition (ii) supposing that the claim amounts are dependent random variables. The observed claims are usually correlated because they contain a common random factor. This kind of correlation can be presented by the following model: For i = 1, 2, . . ., define Yi = J(Ui, V ), where U1, U2, . . . and V are independent r.v.’s, and Ui ∼ F (u), V ∼ G(v), and J(u, v) is some correlated link function. Therefore, the distribution of Yi’s is determined by the link function J(u, v) and two distribution functions F (u) andG(v). A simple example of the correlated link function is the linear function J(u, v) = au+ bv + c. It can be seen that the r.v.’s Y1, Y2, . . . are independent if V is a constant. Our analysis is based on the correlation coefficient, despite its pitfalls. A basic object is to investigate the effect of correlation on the random sum (1). For simplicity, let {Yi} be a sequence of non-negative integer-valued r.v.’s with distribution P (Y = r) = πr, r = 0, 1, 2, . . . , k, ∑k r=0 πr = 1. At first, we consider the following model.