Let 1 , 2 , be independent r v s having an exponential distribution, and E i 1 Let S 0 0, St 1 t , and X t C t exp S t , where C t is a constant such that X t is a martingale (a) Find C t (b) Find lim t X t How fast is such a convergence

Question: Let 1 , 2 , ... be independent r.v.s having an exponential distribution, and E{ i } = 1. Let S 0

Let ξ₁, ξ₂, ... be independent r.v.’s having an exponential distribution, and E{ξ_i} = 1. Let S₀ = 0, St = ξ₁ +...+ξ_t, and X_t = C_t exp{−S_t}, where C_t is a constant such that X_t is a martingale.

(a) Find C_t.

(b) Find lim_t→∞ X_t. How fast is such a convergence?

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