Can Unlimited Sampling Reveal the True Distribution in Statistics?

[Obie]

Suppose that I can sample from some unknown continuous distribution. I know that the draws are iid, but the distribution itself is unknown. However, I know that the true distribution is one of two, either f(x|H) and f(x|L) with common support.

I form the likelihood ratio,

L(x1,...,xN)=[f(x1|H)f(x2|H)...f(xN|H)]/[f(x1|L)f(x2|L)...f(xN|L)]

Is it true that if I can sample an unlimited number of times that I will learn the true distribution for certain? That is, does L(x1,...,xN) converge to zero or infinity in probability (or almost surely)?

For Bernoulli trials, the proof is not hard (Based on LLN), but I am wondering whether this result holds more generally..

 

[Valkarie]

Yes, it is true that if you sample an unlimited number of times, you will learn the true distribution for certain. This result holds more generally because it can be shown that the likelihood ratio converges to either zero or infinity in probability (or almost surely). This follows from the Law of Large Numbers, which states that the average of the results obtained from a large number of trials should be close to the expected value, and the expected value of L(x1,...,xN) is either zero or infinity depending on which distribution is true. Therefore, as the number of trials increases, the likelihood ratio should converge to either zero or infinity in probability (or almost surely).