I'm looking for a reference on random walks on free groups that gives some kind of theorem about existence of a limiting word. A specific case I'm interested in is the free group on two letter $a,b$, where $a,b,a^{-1},b^{-1}$ are all chosen with equal probabilities.
That is, we start with the empty word and at each time step multiply by one of the choices and start building up a word, e.g. $$ aaaba^{-1}bbbab. $$ By "limiting word" I mean the length of the finite word tends to infinity and hence for each $n$, eventually the $n$th letter of the word becomes constant. The limiting word is then defined to be made of these finalized letters.
In the book of David William Probability with Martingales a specific example is given in two exercises (EG.3 and EG.4) but I would like a definitive reference that proves whatever the general version of the theorem I'm talking about is. That is, under what conditions does a limiting word exist?