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Re: Are those variances equal?
Posted:
Jul 1, 2006 8:12 PM


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 popup. >> >>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}.



