MLE Related Proof

Hi,

I am going through the proof in Criag & Hogg (6th edtn) that is the
fundamental result one needs to actually shows that the ML method
"works". There is a step on which I am stuck however, and I am
wondering if anyone can help explain it.

C& H suppose:
x1,...,xn a random sample whose distribution has some pdf f(x;theta)
whose true parameter value is theta = q0.
The regularity conditions are satisfied for the family of distributions
f(x,theta)

They want to prove that for every q ne q0:
Lim n->Inf Prob_q0[ L(q0;x1,..,xn) > L(q;x1,...,xn)] = 1

The step I am getting stuck on is where the use the law of large
numbers. C&H get the result:
1/n * Sum 1 to n { log [f(x1,...,xn;q)/f(x1,...,xn;q0)] } < 0

Then they say "Because the summands are i.i.d. with finite expectation
.... it follows from the law of large numbers that"
1/n * Sum 1 to n { log [f(x1,...,xn;q)/f(x1,...,xn;q0)] } converges
in prob to E_q0{log [f(x1,...,xn;q)/f(x1,...,xn;q0)]}

I am confused by what the random variables _are_. Are C&H talking about
the random variable
log [f(x1,...,xn;q)/f(x1,...,xn;q0)] ? How can they know that this
random variable has a finite mean (since its distribution is not
obvious)?

Thanks for any help you can provide.

JunoE

.

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