I have a Markov chain with transition probabilities $P_{ij}$ and asymtotic distribution $\pi_j$. I'm stuck trying to determine the following limits: $$\lim_{n \to \infty} P(x_n =k, X_{n+1} = j |X_0 = i ) $$ and $$\lim_{n \to \infty} P(x_{n-1} =k, X_{n} = j |X_0 = i ). $$
I don't have experience working with stochastic process. I'm trying to use the Markov property and the definitios of asyptotic distribution without sucess.
Someone can help me?