Talk:Martingale central limit theorem

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There should be made an index of all the central limit theorems which, among others, include this one. --Steffen Grønneberg 20:41, 21 May 2006 (UTC)

[edit] something unclear

There is something not clear :

\tau_v is define as a min. But \sigma_t is a random variable for all t, so the definition is ambiguous. Also the conclusion of the theorem seems a little too good, because there are at the end no condition on the conditional variance. On the other hand, I did not find the theorem in the book you mention as reference. —Preceding unsigned comment added by Le huve (talk • contribs) 12:39, 17 September 2007 (UTC)

Let D_t = X_{t+1} - X_t \,. Then X_1, D_1, \dots D_{t-1} \, can be calculated from X_1, X_2, \dots X_t \, and vice versa. Specifically, X_t = X_1 + D_1 + \dots + D_{t-1} \,. The conditions become:
\operatorname{E}[D_t \vert X_1, D_1, \dots D_{t-1}] = 0 \,
and
|D_t| \le k.
We have
\sigma_t^2 = \operatorname{E}[D_t^2 \vert X_1, D_1, \ldots, D_{t-1}] .
The conditionality of these formulas merely means that each of the random variables D being summed may have a distribution which depends on the values of the previous random variables. Remember we are trying to relax the premises of the central limit theorem. There is no conditionality in the conclusion; we are not trying to weaken the conclusion. JRSpriggs (talk) 00:28, 22 May 2008 (UTC)

[edit] another thing is unclear

Then
\frac{X_{\tau_v}}{\sqrt{v}}
converges in distribution to the normal distribution with mean 0 and variance 1.

Does that mean as v → ∞? Michael Hardy (talk) 19:01, 20 May 2008 (UTC)

Yes, as I indicated in my edit to the article. JRSpriggs (talk) 00:31, 22 May 2008 (UTC)