Limiting distribution and initial distribution of a Markov chain For a Markov chain (can the following discussion be for either discrete time or continuous time, or just discrete time?), 


*

*if for an initial distribution i.e. the distribution of $X_0$, there
exists a limiting distribution for the distribution of $X_t$  as $t
    \to \infty$, I wonder if there exists a limiting distribution for
the distribution of $X_t$  as $t \to \infty$, regardless of the
distribution of $X_0$?

*When talking about limiting distribution of a Markov chain, is it in
the sense that some distributions converge to a distribution? How is
the convergence defined?


Thanks! 

*

*No, let $X$ be a Markov process having each state being absorbing, i.e. if you start from $x$ then you always stay there. For any initial distribution $\delta_x$, there is a limiting distribution which is also $\delta_x$ - but this distribution is different for all initial conditions.

*The convergence of distributions of Markov Chains is usually discussed in terms of 
$$
 \lim_{t\to\infty}\|\nu P_t - \pi\| = 0
$$
where $\nu$ is the initial distribution and $\pi$ is the limiting one, here $\|\cdot\|$ is the   total variation norm. AFAIK there is at least a strong theory for the discrete-time case, see e.g. the book by S. Meyn and R. Tweedie "Markov Chains and Stochastic Stability" - the first edition you can easily find online. In fact, there are also extension of this theory by the same authors to the continuous time case - just check out their work to start with.
