I wonder if the probability kernels of Markov processes on the same state space are close enough, does it also hold for the probabilities of the event that depend only on first $n$ values of the process.
More formally, let $(E,\mathscr E)$ be a measurable space and put $(E^n,\mathscr E^n)$, where $\mathscr E^n$ is the product $\sigma$-algebra. We say that $P$ is a stochastic kernel on $E$ if $ P:E\times\mathscr E\to [0,1] $ is such that $P(x,\cdot)$ is a probability measure on $(E,\mathscr E)$ for all $x\in E$ and $x\mapsto P(\cdot,A)$ is a measurable function for all $A\in \mathscr E$. On the space $b\mathscr E$ of real-valued bounded measurable functions with a sup-norm $\|f\| = \sup\limits_{x\in E}|f(x)|$ we define the operator $ Pf(x) = \int\limits_E f(y)P(x,dy). $ Its induced norm is given by $\|P\| = \sup\limits_{f\in b\mathscr E\setminus 0}\frac{\|Pf\|}{\|f\|}.$ Furthermore, for any stochastic kernel $P$ we can assign the family of probability measures $(\mathsf P_x)_{x\in E}$ on $(E^n,\mathscr E^n)$ which is defined uniquely by $ \mathsf P_x(A_0\times A_1\times \dots\times A_n) = 1_{A_0}(x)\int\limits_{A_n}\dots \int\limits_{A_1}P(x,dx_1)\dots P(x_{n-1},dx_n). $
Let us consider another kernel $\tilde P$ which as well defines the operator on $b\mathscr E$ and the family of probability measures $\tilde{\mathsf P}_x$ on $(E^n,\mathscr E^n)$. I wonder what is the upper-bound on $ \sup\limits_{x\in E}\sup\limits_{F\in \mathscr E^n}|\tilde{\mathsf P}_x(F) - \mathsf P_x(F)|. $
By induction it is easy to prove that $ \sup\limits_{x\in E}|\tilde{\mathsf P}_x(A_0\times A_1\times \dots\times A_n)-\mathsf P_x(A_0\times A_1\times \dots\times A_n)|\leq n\cdot\|\tilde P - P\| $ but I am not sure if this result can be extended to any subset of $\mathscr E^n$.