A stochastic process is said to be **standard Brownian motion** (or a **Wiener process**) if

- almost surely.
- has independent increments, i.e. for , the random variables
are independent.

- For ,
- The map is almost surely continuous.

In this post I follow the method in Sidney Resnick’s *Adventures in Stochastic Processes* to construct Brownian motion on using i.i.d. standard normal random variables. The key to this construction is the following lemma:

Lemma 1Suppose and are random variables such thatThen there exists a random variable such that

*Proof:* Define and suppose is independent of . Define by

so that

Then as , we have

so that and are uncorrelated and hence independent (as they are normally distributed).

Now let

be independent random variables with

Let , , and define using so that and are i.i.d. random variables. Suppose

are defined such that

For each , define such that

and the sequence has independent increments. For each and define the processes

and

That is,

and is linear on each interval . Let be the maximum deviation of and on , partitioned by the intervals , that is,

For each we have

where , and so

For we compute

where and is the probability density of . Let , then we have

Since

from the first Borel-Cantelli lemma we have

from which

This implies that the sequence of functions on is Cauchy, as for

Since is complete, we conclude that exists, with probability , and by construction is Brownian motion on .