Wiener Process

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We define a stochastic process $x(t)$ given by

$$x(nT)=\sum _{i=1}^n{x}_i$$

Where $x_i(i=1,\dots ,n)$ are iid RV with expectation $0$ and variance $T$. The Wiener process $b(t)$ can then be defined as the limit

$$b(t)=\underset{T\to \infty }{\lim }x(t)$$

Where

$$\begin{array}{r}E[b(t{)}^2]=Var\left[\sum _{i=1}^n{x}_i\right]=nT=\frac{t}{T}T=t\end{array}$$