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Topic: Are those variances equal?
Replies: 2   Last Post: Jul 1, 2006 8:15 PM

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Stephen Montgomery-Smith

Posts: 2,351
Registered: 12/6/04
Re: Are those variances equal?
Posted: Jul 1, 2006 8:12 PM
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Konrad Viltersten wrote:
>>>>>Suppose you start off with this expression.
>>>>>V[W_t / t^2]
>>>>>and you wish to show that it tends towards 0 as t->oo.
>>>>> it possible to use the fact that
>>>>>lim t->oo (W_t / t) = 0 (a.s.)
>>>>>and do a rewriting as follows?
>>>>>lim t->oo (V[W_t / t]) = V[lim t->oo (W_t / t)]

>>>>2. Convergence a.s. cannot be converted to convergence in variance
>>>>without some kind of "dominated convergence."

>>>Allright, i take that as a "definitely maybe". I'll look into
>>>what condition i have and hopefully something will pop-up.

>>It is quite often possible to make these things work, but it isn't
>>automatic or necessarily straightforward. But, for example, if W_t is a
>>Weiner process, you even have an explicit formula.

> I'll be working with Wiener processes only, so it's a good
> news. Just to be very clear - the formula you mentioned
> above for Wiener processes is this one, right?
> lim t->oo (V[W_t / t]) = V[lim t->oo (W_t / t)]

No. V[W_t] = t, so V[W_t/t^2] = t/t^4 = t^{-3}.

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