# Continuous time Markov chains and invariance principle

MathOverflow Asked by sharpe on December 6, 2020

This question may be elementary for experts

Let $${xi_n}_{n=1}^{infty}$$ be an i.i.d random variables on a probability space $$(Omega,mathcal{F},P)$$. We assume that the mean of $$xi_n$$ is zero, and the variance is $$1$$. For $$n in mathbb{N}$$ and $$t ge 0$$, we set
begin{align*} X_n&=sum_{k=1}^{n} xi_k,quad Y_t=X_{[t]}+(t-[t])xi_{[t]+1} end{align*}
Here, $$[cdot]$$ denotes the floor function. By the definition, $$Y={Y_t}_{t ge 0}$$ is the linear interpolation of the simple random walk $${X_n}_{n=1}^{infty}$$.
We define $$Z_t^{(n)}=Y_{nt}/sqrt{n}$$, Then, each $${Z_{t}^{(n)}}_{t ge 0}$$ induces a probability measure on $$C([0,infty))$$, the space of continuous functions on $$[0,infty).$$ We denote by $$P_n$$ the probability measure. Donsker’s invariance principle states that $${P_n}_{n=1}^{infty}$$ converges to the Wiener measure.

My question

We write $$n^{-1}mathbb{Z}={cdots,-2/n,-1/n,0,1/n,2/n,cdots}$$. Let $$S^{(n)}={S_t^{(n)}}_{t ge 0}$$ be a (continuous time) Markov chain on $$n^{-1}mathbb{Z}$$. My question is the following:

Are there "Donsker’s invariance principles" for $${S^{(n)}}_{n=1}^{infty}$$ (or scaled $${S^{(n)}}_{n=1}^{infty}$$) ?

Because $${S^{(n)}}_{n=1}^{infty}$$ are continuous time Markov chains, there is no interpolated process like $$Y$$.
Although the state space of each $$S^{(n)}$$ is $$n^{-1}mathbb{Z}$$, this should be regarded as a jump process on $$mathbb{R}$$. Then, each $$S^{(n)}$$ induces an probability measure on $$D([0,infty))$$, the space of right continuous functions on $$[0,infty)$$ with finite left limits. Wiener measures are also regarded as a probability measure on $$D([0,infty))$$.

Please let me know if you have any preceding results.