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Could anybody give me some hints on the following question? I was doing some exercises on Brownian motion and found this online:

Let $\left \{ X_n \right \}_{n=1}^\infty$ be a sequence of independent $N(0,1)$ distributed random variables, defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Also, suppose that $\left \{ \xi_n\right\}_{n=1}^\infty$ is an orthonormal basis for $L^2([0,\infty),\mathcal{B}([0,\infty),\mu)$, where $\mu$ denotes the Lebesgue measure on $[0,\infty)$.

Set $Y_t^{(k)}=\sum_{n=1}^kX_n\int_0^\infty\xi_n(u)\mathbb{1}_{[0,t]}(u)d\mu(u)$, $k\geq1$.

Prove that , for every $t\in[0,\infty)$, the sequence $\left\{ Y_t^{(k)} \right\}_{k=1}^\infty$ converges to in $L^2(\mathbb{P})$ to a random variable $Y_t\in L^2(\mathbb{P})$.

Sorry for the bad format since I know little on Latex. Thanks a lot for your help.

Yiorgos S. Smyrlis
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kurtsThom
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  • Welcome to Math.SE. I suggest you type the question part (c) in MathJax and remove the scan. This shows your dedication and will motivate others to answer it. – AlexR Feb 02 '14 at 22:07
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    @ AlexR: Thanks for your advice. I will edit on it. – kurtsThom Feb 02 '14 at 22:48

1 Answers1

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For (i), we have to prove that the sequence $\{Y_t^{(k)}\}$ is Cauchy because $\mathbb L^2$ is complete. Notice that for $k\geqslant l$ fixed, $$\mathbb E[(Y_t^{(k)}-Y_t^{(l)})^2]=\sum_{n=l+1}^k\left(\int_0^{+\infty}\xi_n(u)\chi_{[0,t]}(u)\mathrm du\right)^2.$$

Notice that if $H$ is Hilbert space, $(e_n)_n$ an orthonormal sequence, then $\sum_{n=1}^{+\infty}\langle x,e_n\rangle^2$ is convergent for any $x$.

Davide Giraudo
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