Random walk on a group

Remarks

The sequence $(Z_n)$ is a Markov chain on $G$ with transition probabilities $p(g, h) = \mu(g^{-1} h)$. These satisfy $p(kg, kh) = p(g, h)$ for all $k \in G$. The distribution of $Z_n$ is the $n$-fold convolution $\mu^{*n}$.

Example: on $\Z^d$, written additively, let $\mu$ give mass $\frac{1}{2d}$ to each of the $2d$ vectors $\pm e_1, \dots, \pm e_d$. Then $Z_n = X_1 + \cdots + X_n$ is the simple random walk on the lattice $\Z^d$, which is the walk on the Cayley graph for the standard generators.

Non-example: on $\Z$, let $X_1$ be $\pm 1$ with probability $\frac{1}{2}$ each, and let $X_n = X_1$ for all $n$. Each $X_n$ has the same distribution, but they are not independent, so $Z_n = n X_1$ is not a random walk on $\Z$ in this sense.