# selection for variance reduction in monte-carlo

I need to prove that the following variance reduction operations lead to the optimum value. A related question i asked here about variance reduction gives the optimal proportions are according to $\sigma_i$. Now i have a selection algorithm that decides which variable to simulate so that maximum reduction is achieved. All variables start from 1 simulation, and then the following selection operation is done before each simulation. $$Initialize\text{ } N_i = 1\\ k=\operatorname*{arg\,max}_i \frac {\sigma_i^2}{N_i(N_i+1)}\\ N_k = N_k + 1 \\ \text{Repeat selection process}\\$$ Prove that this series of operations will lead to the same proportion as the optimum i.e. according to \sigma_i proportions. See my previous question for proof. Is it possible to prove it with infinite series? I have sort of proved it taking advantage of monotonic nature of variance redcution, but i am not satisfied and i wonder if there is a 'strong' proof. I also simulated the selection process and indeed it leads to the same proportions as the optimum. Here is a matlab code to test the idea. The final proportions will be according to sigma even though n started from 1 for all 5 variables. How can i prove this ?

\begin{lstlisting} sigma = [32 16 8 4 2 1]; n = [1 1 1 1 1 1]; for i=1:10000 maxr=0; maxj=1; for j=1:6 t=(sigma(j)^2)/(n(j)*(n(j)+1)); if(t>maxr) maxr=t; maxj=j; end end n(maxj)=n(maxj)+1; n end \end{lstlisting}

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