Abstract

Let W = X<inf>1</inf> + X<inf>2</inf> + · · · + X<inf>N</inf> be a random sum and Z be the standard normal random variable. In this paper, we investigated uniform and non-uniform bounds of the stop-loss distance, which measures the difference between two random variables, W and Z, using the expression |Eh<inf>k</inf> (W) − Eh<inf>k</inf> (Z)|, where h<inf>k</inf> (x) = (x − k)⁺ is a call function. In particular, we focused on the case that X<inf>1</inf>, X<inf>2</inf>, … are independent random variables, and N is a non-negative, integer-valued random variable independent of the X<inf>j</inf> ’s. Our methods were Stein’s method and the concentration inequality approach. The value Eh<inf>k</inf> (W) = E(W − k)⁺ represents the excess over a threshold and is relevant to applications in collateralized debt obligations (CDOs) and the collective risk model.