There exist two types of commonly studied stochastic elliptic models in literature: (I) −∇ · (a x, ω)∇u x, ω)) = f(x) and (II) -∇·(a x,ω)♢∇u x,ω))="f(x), where ω indicates randomness, ♢ the Wick product, and a x, ω) is a positive random process.
More interesting assumptions for the insurance risk problem are that the number of claims N t) is a Poisson process and the sizes of the claims V1, V2,… are independent, identically distributed positive random variables.
positive random variables.
positive random variables with and.
Recall Markov's inequality for a positive random variable (7).
positive random variables with cumulative distribution function (c.d.f).f
We first give an ASCLT for strongly mixing positive random variables.
It allows to approximate a sum of positive random variables by a Gamma distribution.
Therefore, the decay rate k is in fact a positive random variable.
We give here an almost sure central limit theorem for product of sums of strongly mixing positive random variables.
Such models are of the compound type, namely random sums of independent positive random variables, and a reliable statistical model is available for both.
