Random Walk Definition

Question

I have just begun studying this script about Random Walks, but I'm having trouble with a definition that is given there right at the beginning (page 10).

We're looking at Random Walks on the square lattice $\mathbb{Z}^d$ and the definition starts by establishing the notion of a generating set $V=\{x_1, \dots, x_m\}\subset \mathbb{Z}^d$ such that every $y \in \mathbb{Z}^d$ can be written as $y=a_1x_1+\dots+a_mx_m$ for sone $a_1, \dots, a_m \in \mathbb Z^d$. Next, we restrict ourselves to such generating sets $V$ such that for each $x \in V$ the first nonzero component of $x$ is positive. Given such a generating set $V=\{x_1, \dots, x_m\}$ and a function $\kappa: V \rightarrow (0,1]$ with $\kappa(x_1)+\dots+\kappa(x_m) \leq1$ an associated probability distribution $p$ on $\mathbb Z^d$ is defined as follows $$p(x_k)=p(-x_k)=\frac{1}{2}\kappa(x_k),\ p(0)=1-\sum_{x \in V} \kappa(x).$$

My problem: I don't see how $p$ is a well-defined probability distribution on $\mathbb Z^d$. I mean, for an arbitrary $y \in \mathbb Z^d$ we have $p(y)=p(a_1x_1+\dots+a_mx_m)$, but how do I come up with the probability of $y$? It's not like $p$ is a linear function or something, so I guess I'm missing some fundamental point of this definition...

Can anyone help me clarify this?

Answer 1

It is implied (and it should have been said) that all other probabilities are zero, i.e., the probability is supported by the origin, the basis vectors and their opposites; $p(y)=0$ unless $y$ is one of the $x_k$ or its opposite or the origin.

The interpretation is that a random walk has probability zero of taking two jumps (including diagonal jumps) at a time, and that it has equal probability of moving in either of two opposite directions.

Answer 2

As far as I can see, $p$ is just a distribution of a i.i.d random variable $X_i$ taking values on $\mathbb{Z}^d$. With respect to chosen generating set, any member of $\mathbb{Z}^d$ is represented as a coordinate vector $(k_1,\cdots,k_l)$ (using notation used in the original article. Now $p$ is saying that

$P(X_i=(0,0,\cdots,0,\pm 1,0,\cdots,0)) = \kappa(x_k)/2$ with $1$ ocurring at $k^{\text{th}}$ place, and $X_i=0$ with probability $1-\sum_{x\in V}\kappa(x)$ so that $p$ is a legitimate distribution.

So the random walk can be described by $S_n=x+X_1+\cdots+X_n$ where $x$ is the starting position. Probably most familiar example of this would be $x_i=e_i$ where $e_i$ are standard basis, with $\kappa(x_i)=1/d$

But I'm not so sure why we require first non-zero coordinates of generating sets to be positive - I guess author might use this property to imply a certain properties in the random walk, or there's a delicate point that I'm missing here.

But hopefully this kind of gives you the idea what is being described.