Why can one write $\mathbb{P}(f_{j+i}=m|f_i=l) =\mathbb{P}(f_{j+i}=m|f_i=l, f_0=k)$ for Markov Chain?

Question

Why can one write $\mathbb{P}(f_{j+i}=m|f_i=l) =\mathbb{P}(f_{j+i}=m|f_i=l, f_0=k)$ for Markov Chain?

Is this application of Markov property?

Answer 1

Yes. Markov property implies $P(f_{j+i}=m|f_i=l)=P(f_{j+i}=m|f_i=l, f_{i-1}=l_1,...,f_{1}=l_i,f_0=l_{i+1})$ for all choices of $l_1,...,l_{i+1}$ and this gives the desired equality by multiplying by the probability of $f_{i-1}=l_1,...,f_{1}=l_i,f_0=l_{i+1} $ and summing over all values of $l_1,l_2,...,l_i$.