I am trying to understand the intuition that one should get with the Strong Markov Property. Stated by Norris J.R in Markov Chains:
Strong Markov Property: Let $\left(X_{n}\right)_{n>0}$ be Markov$({\alpha}, P)$ and let $T$ be a stopping time w.r.t. $\left(X_{n}\right)_{n \leq 0 .}$ Given that $T<+\infty$ and $X_{T}=i,$ then $\left(X_{T+n}\right)_{n \geq 0}$ is a Markov chain $\left(\delta_{i}, P\right)$ which is independent of the past $X_{0}, X_{1}, \ldots, X_{T-1}$
If I understood correctly, does that mean that if we start to look at a sequence $X_n$ of random variables from a time given $T$, it is still a Markov Chain? Also why do we have a Kronecker delta in the statement?
A Markov Chain is described by an initial distribution $\lambda$ and a transition matrix $P$. What the strong Markov property tells you is that the behaviour of process $(X_n)$ starts afresh from $i$ at the random time $T$. Since we know the state at that time $T$ and it is $X_T=i$, the intial distribution is simply the unit mass distribution at $i$ and that is indicated by $\delta_i$ .