Can Strong LLN and Weak LLN apply to continuous distributions?

Question

Can Strong LLN and Weak LLN apply to continuous distributions? Or it can only apply to discrete distributions?

Representation of LLN: $X_1,X_2,\ldots,X_n$ are i.i.d, and their expectation values are finite, then $$(X_1+X_2+\cdots+X_n)/n \longrightarrow \mathrm{E}(X),$$ as $n$ goes to infinity.

Moreover, just to confirm, Central limit theorem also can apply to continuous distributions, right?

Answer 1

Quoting from Wikipedia

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed

As you can deduce from the above quote, there is no mention of "discrete r.v. only" whatsoever. The WLLN, which is a particular example of convergence in probability, applies to every sequence of r.v.'s for which the moment conditions are satisfied. Indeed, the classical proof of the WLLN is based on the Chebyshev inequality, which holds for any r.v.

However, the WLLN fails when the required moment conditions are not satisfied. A notable example of such a failure is when $X_i$ are Cauchy r.v.'s.

The Central Limit Theorem (CLT) is a particular example of convergence in distribution. The CLT (like the WLLN) applies to any sequence of r.v.'s that satisfies the required moment conditions.