In Stochastic Simulation: Algorithms and Analysis by Glynn and Asmussen on p 146 they provide the following example.

Let $f(x)=a/(1+x)^{a+1}$ be the density of a pareto distribution and let $a=3/2$ (scale = 1). We can then estimate $f^{*n}$ (the convolution density, i.e. density of $X_1,\dots,X_n$ i.i.d. pareto distributed) by using conditional monte carlo. This uses the observation that conditional on $S_{n-1}=\sum_{i=1}^{n-1}X_i$, $S_n$ should have density $f(x-S_{n-1})$. Furthermore by the law of large numbers we should be able to simulate $R\cdot n$ paretos, and for each set $X_1^i,\dots,X_n^i$ $i\in\{1,\dots,R\}$ we get an conditional estimate of $f^{*n}$, taking the empirical average over $R$ should yield an estimate of $f^{*n}$.

First of all, are the above an correct understanding? Forr those of you who don't have the book I hope my description is thorough enough so that you can say from that alone. Second they don't mention anything about doing something special if $x<S_{n-1}$, but I'm thinking we have to exclude those $x$ (It's the only idea I have). Is that correct or what do you do with those? If my idea is correct how do we get any information of small $x$? Just hope for low $S_{n-1}$?


p.s. I hope this is the right stackexchange else, please let me know.


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