Probability distribution on finite group I'm preparing for finals and this is a practice question. I'm not really sure how to start, so any solutions/hints/starting points are appreciated. 
Suppose $P = \sum a_gg$ were a probability distribution on a finite group $G$. If $a_1 > 0$, then $\lim_{k\rightarrow \infty} P^k$ exists and is equal to ${1\over|H|}\sum_{h\in H}a_hh$, where $H$ is the subgroup of $G$ generated by elements with $a_g > 0$.

The fact that this is tagged representation-theory makes me feel like the following isn't what you were looking for, but here it goes anyway...
The element $P$ describes a Markov chain on $H$, where $a_h$ is the transition probability between $x$ and $xh$, for all $x,h \in H$.  It is easy to check that $P^k$ is the probability distribution over $H$ after $k$ steps of the Markov chain.  Since $a_1 > 0$, the Markov chain is aperiodic, and since every element of $H$ can be expressed a product of $a_g$ with $a_g > 0$, the Markov chain is irreducible.  Now by standard results in the theory of Markov chains, an irreducible aperiodic Markov chain has a unique stationary distribution $S = \lim_{k \to \infty} P^k$, and this stationary distribution is given by the unique solution to the equation $PS=S$ such that $S$ is a probability distribution.  It is easy to check that $S = \frac{1}{|H|}\sum_{h \in H} h$ is a probability distribution and satisfies $PS=S$.
