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Topic: Are those variances equal?
Replies: 1   Last Post: Jun 30, 2006 7:19 PM

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Stephen Montgomery-Smith

Posts: 2,351
Registered: 12/6/04
Re: Are those variances equal?
Posted: Jun 30, 2006 7:19 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.
> What we tried is this rephrasal.
> (1/t) * V[W_t / t]
> Does it hold?
> And then, 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)]

1. V(a W_t) = a^2 V(W_t) for any scalar a.

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

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