Question: Exercise 11.20 If N is a Poisson process with rate , show that the moment generating function of Ut = Nt E(Nt ) var(Nt

Exercise 11.20 If N is a Poisson process with rate λ, show that the moment generating function of Ut =

Nt − E(Nt )

√var(Nt )

is Mt (x) = E(exUt ) = exp



−x√λt + λt (ex/√λt − 1)



.

Deduce that, as t →∞, P(Ut ≤ u) →

Z u

−∞

1

√2π

e−1 2 v2 dv for u ∈ R.

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