By K. Mathiak, Steffen Supra

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For example, the number of claims exceeding a high threshold u, say, in the period (a, b] of time is given by M ((u, ∞) × (a, b]) = #{i ≥ 1 : Xi > u , Ti ∈ (a, b]} . This is a Pois((1−F (u)) µ(a, b]) distributed random variable. It is independent of the number of claims below the threshold u occurring in the same time interval. Indeed, the sets (u, ∞) × (a, b] and [0, u] × (a, b] are disjoint and therefore M ((u, ∞) × (a, b]) and M ([0, u] × (a, b]) are independent Poisson distributed random variables.

The estimates ﬂuctuate around the value 1. Bottom right: QQ-plot of the values µ(Tn ) − µ(Tn−1 ) against the standard exponential distribution. The plot curves down at the right end indicating that the values come from a distribution with tails heavier than exponential. 19. Top left: The cember 31, 1980. The inter-arrival times have sample mean λ renewal process N (t) generated by the arrivals (solid boldface curve). 19)−1 is drawn. Top right: The histogram of the inter-arrival times. For comparison, the density of the Exp(λ) distribution is drawn.

E. unique probability density of (X(1) , . . 11 We are now ready to formulate one of the main results of this course. e. s. Then the conditional distribution of (T1 , . . , Tn ) given {N (t) = n} is the distribution of the ordered sample (X(1) , . . , X(n) ) of an iid sample X1 , . . , Xn with common density λ(x)/µ(t), 0 < x ≤ t : d (T1 , . . , Tn | N (t) = n) = (X(1) , . . , X(n) ) . ,Tn (x1 , . . , xn | N (t) = n) = n! 15) i=1 0 < x1 < · · · < xn < t . Proof. We show that the limit P (T1 ∈ (x1 , x1 + h1 ] , .