| The Asymptotic Behavior in the Renewal Model |
| Written by theoretic |
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In this section we are interested in the asymptotic behavior of the total claim amount process. Throughout we assume the renewal model (see p. 77) for the total claim amount process S. As a matter of fact, S(t) satisfies quite a general strong law of large numbers and central limit theorem: Figure 3.1.4 Visualization of the strong law of large numbers for the total claim amount S in the Cramer-Lundberg model with unit Poisson intensity. Five sample paths of the process (S(t)/t) are drawn in the interval [0,1000]. Left: Standard exponential claim sizes. Right: Pareto distributed claim sizes Xi = 1 + (Yi -EYi)Ipvar(Yi) for iid Yi’s with distribution function P(Yi < x) = 1 - 24x~4, x > 2. These random variables have mean and variance 1. The fluctuations of S(t)/t around the mean 1 for small t are more pronounced than for exponential claim sizes. The right tail of the distribution of X\ is much heavier than the right tail of the exponential distribution. Therefore much larger claim sizes may occur. |