Consider a free group with generators $a$ and $b$. A word comprises a product of symbols, where a symbol is one of $a$, $b$ or their respective inverses, $a^{-1}$ and $b^{-1}$. We denote $1$ as the empty word. A reduction operation $f$ removes sucessive instances of a symbol and its inverse from a word until no more remain, so $f(1a)=a$ and $f(bb^{-1})=1$. Define $(X_n)$ as follows:
- Sample $\xi_n$ uniformly from $\{a,a^{-1},b,b^{-1}\}$
- Set $X_n=f(X_{n-1},\xi_n)$, i.e we multipltly $\xi_n$ on the right of $X_{n-1}$ and the perform reduction via $f$.
Let $g$ be the function counts the number of symbols in a word. $g(a,a,b)=3$, but $g(1)=0$, i.e. the empty word has $0$ length. Set $Y_n:=g(X_n)$.
Define $T:=\inf\{n\geq1:Y_n=5\}$, the question is how to calculate $\mathbb{P}(X_T=a,a,a,b,b)$ starts form $X_0=1$.
In my opinion, this probability is equal to starts from $X_0=a,a,a,b,b$ then the empty words reoccurs. However, the length of $X_n$ should not exceed $5$. So how can I take care of this boundary condition?