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Approximation of random sums of random variables in insurance

Wlaź P.

Advances in Science and Technology. Research Journal

2014

Vol. 8, nr 23

88--93

The paper deals with approximations of random sums. By random sum we mean a sum of random number of independent and identically distributed random variables. Distribution of this sum is called a compound distribution. The model is especially important in non-life insurance. There are many methods for approximating compound distributions, one of the most popular one is approximation with shifted gamma distribution. In this work we show an alternative way – using kernel density, Fast Fourier Transform and numerical optimization methods – for finding shifted gamma approximations and show results suggesting its superiority over classical method.

Random sums of independent random vectors attracted by (semi)-stable hemigroups

Becker-Kern P.

Journal of Applied Analysis

2004

Vol. 10, nr 1

83-104

Let (Xn) be a sequence of independent not necessarily identically distributed random vectors belonging to the domain of attraction of a stable or semistable hemigroup, i.e. for an increasing sampling sequence (kn) such that kn+1/kn → c ≥ 1 and linear operators An, the normalized sums [wzór] converge in distribution uniformly on compact subsets of {0 ≤ s < t} to some full probability μs,t. Suppose that (Tn) is a sequence of positive integer valued random variables such that Tn/kn converges in probability to some positive random variable, where we do not assume (Xn) and (Tn) to be independent. Then weak limit theorems of random sums, where the sampling sequence (kn) is replaced by random sample sizes (Tn), are presented.