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:

Kallenberg

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 …