Let $W_n$ is sequence of random variables. Is following condition $$\mathbb{E}\;W_n^2 \to c$$

sufficient for $W_n$ to converge in $L^2$-norm i.e. $\exists W\in L^2(\mathbb{P})$ such that $\; \mathbb{E}(W_n-W)^2\to 0,;n\to \infty$.

I was reading book about Brownian motion from Schilling. In chapter about consturction of Brownian motion, he creates sequence

$W_N(t)= \sum_{n=1}^{N-1} G_n \int_{0}^t \phi_n(s)ds$

where $G_n$ are iid $\mathcal{N}(0,1)$ for every $t\in [0,1]$.

He wants to prove that $W_n(t)$ converges in $L^2$ for every $t$. He only prove $\mathbb{E} W_N^2(t)$ converges to $0$ and state that $L^2 – \lim_{N\to \infty} W_N(t)$ exists. Can someone explain me how did he get it?

## Best Answer

In general, condition $\mathbb E[W_n^2] \to c$ for some $c \in \mathbb R$ isn't sufficient to state that $W_n \to W$ in $L_2$ for some random variable $W$. Indeed, take $(\Omega,\mathcal F,\mathbb P)= ([0,1],\mathcal B([0,1]),\mathcal L)$ where $\mathcal L$ is Lebesgue measure on $[0,1]$. Define $W_n = \sqrt{n}1_{[0,\frac{1}{n}]}$. Then $\mathbb E[W_n^2] = n\mathcal L([0,\frac{1}{n}]) = 1 \xrightarrow[n \to \infty]{} 1$. Note that $W_n \to 0$ almost surely, hence in probability, too. If there is some $W$ such that $W_n \to W$ in $L_2$, then $W_n \to W$ in probability, too. Since limit in probability is unique almost surely, we would get that $W=0$ almost surely. But $\mathbb E[(W_n-0)^2] = \mathbb E[W_n^2] = 1$ which doesn't converge to $0$.

For the second example, if we have $W_n$ in the special form, that is $W_n(t) = \sum_{k=1}^n G_k f_k(t)$ for some deterministic functions $f_k$ and family $\{G_k\}_k$ of i.i.d standard normal, and we know that $\mathbb E[W_n^2(t)] = \sum_{k=1}^n f_k^2(t) \to \sum_{k=1}^\infty f_k^2(t) < \infty$ then it is enough to claim that $W_n$ converges in $L_2$ to some limit $W$. Indeed, note that $W_n$ is Cauchy in $L_2$, because for $m > n$ we have $$ \mathbb E[|W_m(t)-W_n(t)|^2] = \sum_{k=n+1}^m f_k^2(t) \le \sum_{k=n+1}^\infty f_k^2(t) \xrightarrow[n \to \infty]{} 0 $$ since the latter is a tail of convergent series. So that $(W_n(t))_n$ is Cauchy in $L_2$. Since $L_2$ is a complete space, then $(W_n(t))_n$ is convergent, which means that there is some $W(t) \in L_2$ such that $W_n(t) \to W(t)$ in $L_2$

In your example $f_k(t) = \int_0^t \phi_k(s)ds$, which by Parseval theorem and the fact that $\{\phi_n\}_n$ forms an orthonormal basis, gives $\sum_{k=1}^\infty f_k^2(t) = t < \infty$.