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I am attempting to capture what I am doing when running a simulation, drawing discrete random samples from a normal distribution.

Assume $X \sim \mathcal N(\mu,\sigma^2)$. I then draw $n$ samples $S_i$ from the continuous sample space $\Omega$.

Does the following make sense, and if so, does it explain what I am doing?

$\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \mathbb{1}_{Z=i} S_i \rightarrow X$,

with $\mathbb{1}_{Z=i}$ the indicator function and $Z$ the multinomial,

$ Z \sim \mbox{Mult}(n;1/n)$.

What confuses me is that I have a countable set approaching an uncountable set as $n \rightarrow \infty$.

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On a more practical solution view relating to your question, I suggest the following paper: A Sample Approximation Approach for Optimization with Probabilistic Constraints available here, to quote from the abstract:

We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with risk level larger than the required risk level will yield a lower bound to the true optimal value with probability approaching one exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence that the sample approximation will yield a lower bound. We then provide conditions under which solving a sample approximation problem with a risk level smaller than the required risk level will yield feasible solutions to the original problem with high probability. Once again, we obtain a priori estimates on the sample size required to obtain high confidence that the sample approximation problem will yield a feasible solution to the original problem. Finally, we present numerical illustrations of how these results can be used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic constraints.

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  • $\begingroup$ Thank you for the comment and reference @AJKOER. As I understand the article, it shows that with enough samples you can get suitably close to the truth. (My upvote does not show as I am a new member.) $\endgroup$
    – GCru
    Sep 20, 2021 at 7:37

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