What is the notation for a random walk function?

Question

I would like to define a 1d random walk function, shall I write something like:

def: $ x(t) = random(t)$ ?

What is the proper mathematical notation for random walk function?

Math newbie thanks your help.

Answer 1

Unfortunately, in order to define a random walk you need some more rigorous mathematics. It is not only about the notation. We distinguish discrete and continuous time random walks. Let me start with a definition of the simple random walk.

Discrete time

The simple random walk

Let $X_1, X_2, \ldots,$ be a sequence of independent random variables such that each random variable is Bernoulli with support $\{-1,1\}$, that is $$X_k = \begin{cases} 1 & \mbox{with probability } p \in [0,1]\\ -1 &\mbox{with probability } 1- p\end{cases}$$ for all $k \in \mathbb{N}$, or you may prefer to write $\mathbf{Pr}(X_k = 1)= p$, $\mathbf{Pr}(X_k = -1)= 1-p$.

You can think about each $X_k$ as the $k$-th step of a random walker, in one-dimension, she/he can either go right with probability $p$ or left with probability $1-p$. In order to have define a random walk we need to sum up the steps, namely $$S_n = S_0 + \sum_{k=1}^n X_k = S_0 + X_1 + X_2 + \cdots + X_n$$ for all $n \in \mathbb{N}$, where $S_0 \in \mathbb{Z}$ is the initial position (home) of the walker. In order to have the standard symmetric simple random walk we take $S_0 = 0$ and $p=\frac{1}{2}$.

General case

The above random walk is the most intuitive one, however we do not have to be restricted to Bernoulli distribution. We can take a sequence of independent, identically distributed (iid) random variables and still define $S_n$. One reason behind it, is that if we `scale' this random walk (and take each r.v. to have zero expectation and unit variance) and take the limit (in distribution) as $n \to \infty$ we obtain the famous continuous-time random walk, called the Brownian motion (the Wiener process). This result is known as Donsker's invariance principle or the functional central limit theorem.

Continuous-time

Standard Brownian Motion

The famous continuous-time random walk, the standard Brownian motion is a continuous-time stochastic process with a few properties which I will list below. The usual notations are $B=(B_t)_{t \geq 0}$ or $W= (W_t)_{t \geq 0}$, $B$ for Brownian and $W$ for Wiener. For a fixed $t\geq 0$, $B_t$ stands for a position of a random walker at time $t$. Here is the promised list of properties

1. $B_0 = 0$ almost surely.
2. $B$ has the independent increment property, that is, for all $t >0$ the future increments $B_{t+s}-B_t$, $s \geq 0$ are independent of the past $B_r$, $r \leq t$.
3. The increments of $B$ have normal distribution, that is, $B_{t+s}-B_t \sim \mathcal{N}(0,s),$ where $\mathcal{N}(\mu, \sigma^2)$ stands for the normal distribution with mean $\mu$ and variance $\sigma^2$.
4. $B$ has continuous sample-paths, that is, $B_t$ is continuous in $t$ almost surely.

The second property implies that Brownian motion is a Markov process, roughly speaking, the future of the walker depends only on the present, not on the past.

Note that the first and third properties imply that $B_t \sim \mathcal{N}(0,t)$ for all $t \geq 0$. The third property let us quite easily simulate the Brownian motion on the computer, of course we have to discretise the time in order to do that.

The fourth property tells us that our walker does not jump, that is, moves continuously.

Brownian motion is a fundamental stochastic process, it is not a simple notion. Proving its existence is a very non trivial task, there are several different constructions. Perhaps the most intuitive one is the one via discrete-time random walks, which I mentioned earlier. I hope my answer helps.