Conditions in law of large numbers

Question

The (Strong) Law of large numbers states that

$ \frac{1}{N}\sum_{k=1}^N h(X_k) \rightarrow \mathbb{E}\left[h(X)\right]$

a.s in $\mu$ as $N\rightarrow \infty$.

but I can't find any conditions on $h(\cdot)$ other than it needs to be $\mu$-integrable.

I am not overly comfortable with measure and integration theory so I might miss the point here but what conditions are set on $h$? The proofs I have found on LLN that doesn't involve measure theory, doesn't have any function $h$ and I don't really know what is included in being $\mu$-integrable.

I need this in my thesis for explaining why my Sequential Monte Carlo estimators are consistent. Any reference to a book that has this proof might be really helpful but understanding is my biggest priority here.

Answer 1

For iid data, the only condition is that $\mathbb E[h(X)]$ exists. If $\mathbb E[h(X)] = +\infty$ or $-\infty$ the SLLN still holds; the only issue comes when $\mathbb E[h(X)]$ is undefined, i.e. the positive and negative parts of $h(X)$ integrate to $\infty$. The integrability condition (which is overly strong for iid data) is essentially saying that $\int |h(X)| \ d\mu < \infty$ where $(\Omega, \mathcal F, \mu)$ is some underlying probability space; usually one would check that $\int |h(x)| f(x) < \infty$ if $X$ is continuous or that $\sum |h(x)| f(x) < \infty$ if $X$ is discrete where $f$ is the pdf/pmf of $X$.

This is all assuming iid data; for independent data, or for Markov chains, things are more complicated. With Markov chains, some measure theory is probably inescapable.