Let $W=(G,S,P)$ be a directed random walk. Then the probability distribution $Q^k(g)$ tends to uniform stationary distribution $\pi(g)=\frac{1}{|G|}$ as $k$ tends to infinity.
Consider an element $g=s_1s_2...s_k \in G$, where $k$ is as small as possible. Then the probability of getting to $g$ starting from the identity element in $n$ steps can be calculated in two cases: Suppose $n>k$.
Here then probability of reaching to $g$ is $P(s_1)P(s_2)...P(s_k)*EE$ where $EE$ is the probability to reaching $e$ in $n-k$ steps starting at $e$.
we essentially need to prove that $EE$ equals to $\frac{1}{|G|*P(s_1)P(s_2)...P(s_k)}$ Because then when plugging this in the place of $EE$ we'll get that $Q^k(g)$ tends to uniform distribution. Now I'm stuck.