Extreme-value Process. Claims arrive at an insurance company at times Tn that form a Poisson process N with rate λ. The size Yn of the nth claim that arrives at time Tn has an exponential distribution...

Extreme-value Process. Claims arrive at an insurance company at times Tn that form a Poisson process N with rate λ. The size Yn of the nth claim that arrives at time Tn has an exponential distribution with rate μ and the claim sizes are independent of their arrival times. The maximum claim up to time t is X(t) = maxk≤N(t) Yk. Justify that X(t) is a CTMC and specify its defining parameters. Show that X(t) → ∞ a.s. as t → ∞, and that μX(t) − log(λt) d → Z, where P{Z ≤ x} = exp{−e−x}, which is the Gumbel distribution. Evaluate the distribution by conditioning on N(t) and using the exponential property that if nan → a, then (1 − an)n → e−a as n → ∞. As an intermediate step, justify that μX(T_n)−log(λ_n) d → Z by evaluating the distribution of μX(T_n).