Variance of S(t) in the Cramer-Lundberg and renewal models
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In the Cramer-Lundberg model the Poisson distribution of N(t) gives us EN(t) = var(N(t)) = At. Hence
var(£(£)) = At [var(Xi) + (EXi)2] = At E(X2).
In the renewal model we again depend on some asymptotic formulae for EN(t) and var(N(t)); see Theorem 2.2.7 and Proposition 2.2.10:
var(£(£)) = [At var(Xi) + var(T/l/i) A31 (EXi) 2] (1 + o(1))
= At [var(Xi) + var(T/l/i) A2 (EXi) 2] (1 + o(1)).
We summarize our findings.