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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.
>>>>>...is 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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