Suppose we have two transition probability matrices $\hat{\textbf{P}}$ and $\textbf{P}$. I'm searching for an upper bound on the Frobenius norm of their difference $$\|\hat{\textbf{P}} - \textbf{P}\|_{F}.$$
I can show that the row sums are bounded by 2, which would mean the summation is bounded by $2k$ assuming $k$ is the number of states, and so the norm would be bounded by $\sqrt{2k}$. But it seems like I can do better based on test data.