Inductive definition of a random walk?

Question

I am currently looking at some lecture notes in which the following is found:

I have covered random walks previously, and the definition that I am familiar with is as follows:

A random walk is a stochastic process $X$ where $X_n = X_{n-1} + Z_n$, $n \in \mathbb{N}_0$ where the $Z_n$ (known as the increments of the random walk) are independent, identically distributed random variables independent of $X_0$.

My issues with the definition in the lecture notes are as follows:

1. When it says "$W_n(t)$ has steps $\delta = \frac{1}{n}$", does the term "steps" refer to the size of the increments by which the process may change or the frequency with which the process may change value.
2. If the term "steps" refers to the size of the increments in the process, then shouldn't the second formula in the lecture note definition be $$ W_n (k+1) := W_n (k) + \frac{1}{\xi_k} $$ where $\xi_k \in \{ +n, -n \}$
3. If the term "steps" refers to the frequency with which the process changes value, then shouldn't the second formula in the lecture note definition be $$ W_n \left( \frac{k+1}{n} \right) := W_n \left( \frac{k}{n} \right) + \xi_k $$
4. If the second formula in the lecture note definition is indeed correct, then this implies that the frequency with which the process may change value and the size of the increments by which the processes value may change are proportional. That is, it implies that after time $\frac{1}{n}$ the process will change in value by $\frac{1}{\sqrt{n}}$. Why would this be true?

EDIT: I suspect that the definition is just poorly written and is, in fact, describing one particular random walk, rather than giving a general definition. I am still uncertain, however.

Answer 1

The term "step" means "steps in time"... and the time is your arguments in brackets as the notation "$W_n(t)$" already suggest… $t$ stands for for "time" (as usual for stochastic processes, but still good you asked!).

While on a standard random walk the step size is 1 AND the increment size is in $\{-1,1\}$, your increment size here is (as you already figured out yourself) in $\left\{-\frac{1}{\sqrt{n}},\frac{1}{\sqrt{n}}\right\}$

From that point of view your given definition is just a special case of the definition you found. Just be careful and don't mix up the notations! The $n$ in the definition you found is different from that in your definition. Actually your definition defines a SEQUENCE of random walks… namely a for each $n\in\Bbb N$ a new random walk.

But why in the way it does? Your proposal of defining $$W_n \left( \frac{k+1}{n} \right) := W_n \left( \frac{k}{n} \right) + \xi_k$$ without having the factor $\frac{1}{\sqrt{n}}$ defines a random walk as well… but the goal of the definition is to define a sequence that converges in a "nice" sense

You are right that "intuitively" one may start with altering the increments the same way like steps in time, but let's consider the case $n=1$ in you definition, what's a standard random walk then it holds $$\begin{align}W_1(t) = \sum_{k=1}^{\lfloor t\rfloor} \xi_k\end{align}$$.

Now let's calc the expectation and variance and we get:

$$E[W_1(t)] = 0 \\ \text{Var}(W_1(t)) = \lfloor t\rfloor$$

The same way like above we can write $$W_n(t) = \frac{1}{\sqrt{n}}\sum_{k=1}^{\lfloor nt\rfloor} \xi_k$$ and we get

$$\begin{align}E[W_n(t)]& = 0 \\ \text{Var}(W_n(t)) &= \frac{1}{n}\lfloor nt\rfloor\end{align}$$

so for $n\to\infty$ we have $\text{Var}(W_n(t)) \to t$