Let (N(t)),>, be a Poisson process with rate 1, and for e > 0, let X,...

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Let (N(t)),>, be a Poisson process with rate 1, and for e > 0, let X, X,... be i.i.d. with density C'yle-", y > ɛ, where the normalising constant is x-le"dx. Finally, for t > 0, define the time-scaled compound Poisson process N(tCe) %3D j=1 (a) Show that for any ɛ > 0, C; <elee, and that for any 0 < ɛ < 1, el log(1/e) < Ce. (b) Show that the number of jumps N(tCe) of the process (Z20 up to time t converges to infinity in probability as e → 0+. (That is, show that for all n E N, P(N(sC) > n) → 1 as e → 0+.) (c) Show that the Laplace transform of Z): L«(0) := E[e=0z;°], 0 >0, converges pointwise as e → 0+, and identify the limit as the Laplace transform of a well-known distribution. (d) Explain in one or two sentences how the number of jumps can go to infinity, but the distribution of Ze) can converge. In fact, the whole process (Z)t20 converges to a process having independent incre- ments and marginals given by part (c). The limit is a non-decreasing pure jump process, with the times of the jumps dense in the positive line. Let (N(t)),>, be a Poisson process with rate 1, and for e > 0, let X, X,... be i.i.d. with density C'yle-", y > ɛ, where the normalising constant is x-le"dx. Finally, for t > 0, define the time-scaled compound Poisson process N(tCe) %3D j=1 (a) Show that for any ɛ > 0, C; <elee, and that for any 0 < ɛ < 1, el log(1/e) < Ce. (b) Show that the number of jumps N(tCe) of the process (Z20 up to time t converges to infinity in probability as e → 0+. (That is, show that for all n E N, P(N(sC) > n) → 1 as e → 0+.) (c) Show that the Laplace transform of Z): L«(0) := E[e=0z;°], 0 >0, converges pointwise as e → 0+, and identify the limit as the Laplace transform of a well-known distribution. (d) Explain in one or two sentences how the number of jumps can go to infinity, but the distribution of Ze) can converge. In fact, the whole process (Z)t20 converges to a process having independent incre- ments and marginals given by part (c). The limit is a non-decreasing pure jump process, with the times of the jumps dense in the positive line.

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a Let X1 X2 be a sequence of iid random variables with common distribution F For any t 0 define the ... View the full answer