Bounds on stop-loss distance for generalized multinomial random sum

Abstract

Let WN = X1 + X2 + \cdots + XN be a random sum, where N is a non-negative, integer-valued random variable independent of the sequence (Xj). This dissertation studies bounds for the stop-loss distance, |E(WN-k)+ - E(Y-k)+|, where Y follows either a normal or a Poisson distribution. We also present illustrative examples demonstrating that E(WN-k)+ plays a central role in actuarial mathematics and financial risk management. Our work covers both independent and dependent frameworks. To model dependence, we adopt the generalized multinomial (GM) model introduced by Tallis (1962) and later extended by Daly (2022), which incorporates an equicorrelated structure among the Xj's. Within this setting, we establish refined uniform and non-uniform bounds under both normal and Poisson approximations. The methodology is based on Stein’s method, combined with concentration inequalities and biasing techniques.