Consider a hidden markov model (HMM) with two hidden states $A$ and $B$ and emission support $1$ and $2$ fitted with initial state distribution $$\lambda = [\begin{array}{cc} .7&.3\end{array}]$$ with a row-stochastic transition matrix $$T = \left[\begin{array}{cc} .3&.7 \\ .2&.8\end{array}\right]$$ and row-stochastic emission matrix $$O = \left[\begin{array}{cc} .2&.8 \\ .5&.5\end{array}\right].$$
I would like to find the limiting unconditional distribution over the emission values, i.e. the distribution over observations $o_t$ such that $o_t = o_{t-1}.$ We can see that we can calculate $o_1 = \lambda O$, although I am having difficulty find a calculation for $o_t$ where $t > 1.$ Any suggestions?