pr.probability – Convergence of the Markov chain in discrete time to Feller processes

Let

• $$( Omega, mathcal A, operatorname P)$$ to be a probability space
• $$(X_t) _ {t ge0}$$ to be a Feller process on $$( Omega, mathcal A, operatorname P)$$
• $$left (Y ^ {(d)} _ n right) _ {n in mathbb N_0}$$ to be a homogeneous Markov chain over time on $$( Omega, mathcal A, operatorname P)$$ and $$X ^ {(d)} _ t: = Y ^ {(d)} _ { lfloor frac t {h_d} rfloor} ; ; ; text {for} t ge0$$ for $$d in mathbb N$$
• $$(h_d) _ {d in mathbb N} subseteq (0, infty)$$ with $$h_d xrightarrow {n to infty} 0$$
• $$N$$ to be a Poisson process on $$( Omega, mathcal A, operatorname P)$$ with parameter $$1$$ independent of $$Y ^ {(d)}$$ for everyone $$d in mathbb N$$ and $$N ^ {(d)} _ t: = N _ { frac t {h_d}} ; ; ; text {for} t ge0$$ as good as $$tilde X ^ {(d)} _ t: = Y ^ {(d)} _ {N ^ {(d)} _ t} ; ; ; text {for} t ge0$$ for $$d in mathbb N$$

Note that $$N ^ {(d)}$$ is a Poisson process with parameter $$h_d ^ {- 1}$$ for everyone $$d in mathbb N$$.

How can we show that (in the Skorohod topology) $$X ^ {(d)} xrightarrow {d to infty} X$$ Yes Yes $$tilde X ^ {(d)} xrightarrow {d to infty} X$$?

In Kallenberg's book, the author mentions that the statement stems from the following two theorems:

I do not understand how we should apply them. Clearly, for fixed $$t ge0$$we can consider $$frac1d sum_ {i = 1} ^ d left (N ^ {(i)} _t-N ^ {(i-1)} _t right)$$ with $$N ^ {(0)} _ t: = 0$$. However, while being independent, the $$N ^ {(i)} _ t-N ^ {(i-1)}$$ are do not identically distributed …