It's funny.

I just covered this in my lecture today.

This is a lame theorem.

Consistency, actually Weak Consistency is just convergence in Probability.

Strong Consistency is almost sure convergence.

(This is what I do for a living. Study the difference between these two.

For example the St Peterburg game.)

The answer to your first question is NO.

This lame theorem is just Chebyshev's Inequality.

So if the variance goes to zero then theta hat will approach theta in the weak sense.

Now in your two examples, the first one is on page 452 of Wackerly,

which I went over today.

And since the 4th moment is finite S^2 converges in probability to sigma^2.

But you can get a stronger result, I'm just using your theorem.

The one second one has NO limit.

X_1-X_2 is a random variable, you can easily get it's distribution.

There's no sequence here.

So it has no limit.

MOST things do not have a limit.

X_1-X_2 is a Normal with mean 0 and variance 2 sigma^2, by indep,

Then (X_1-X_2)^2 is a mulltiple of a chi-square.

Dividing it by two only means we can get its exact distribution.

It has a spread, it will not converge to anything.

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Here you go.

Convergence of random variables - Wikipedia, the free encyclopedia

I deal with convergence in law/distribution a little bit, for example the central limit theorem.

But mostly I look at convergence in probability and almost sure convergence.

I hardly look at L_p convergence, or convergence in mean.

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You confused me a bit

NO it is not IFF

BUT

Now my question is, if the limit is NOT zero, can we conclude that the estimator is NOT consistent? (i.e. is the theorem actually "if and only if", or is the theorem just one way?)

is confusing

The converse is false here.

BUT the CONTRAPOSITIVE is always true

A implies B

means Not B implies Not A

So yes, if the variance does not go to zero it is not consistent.

BUT that's what I showed.

the (X_1-X_2)^2/2 is a rv it will have spread, hence it variance does NOT

go to zero.

so no and yes to your question.

It is one way, but you mixed up converse and contrapositive.