Standard Statistics Question

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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..
 

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