Limit theorem : reproduce a proof with an adaption from discrete to continuous time

Im considering Theorem 5.2.2 in M. Sørensen "Exponential Families of stochastic processes".

The setup is as follows: We have a Levy-Process $X_t$ fullfilling the CLT \begin{align} \sqrt{t}(X_t/t-E(X_1))\xrightarrow{d}\mathcal{N}(0,\sigma^{2})\qquad (1) \end{align}

Now let $S_t$ be a continuous positive random function of $t$ with $S_t\rightarrow \infty$ a.s. Let $f_t$ be a positive nondecreasing function with $f_t\rightarrow \infty$ such that $S_t/f_t\rightarrow \nu$ a.s. where $\nu$ is a non-negative finite random variable.

Now Sørensen and Küchler state that (1) is also mixing-convergence(definition see below).

After this they say that (1) holds mixing even under a random change of time $S_{t}$ instead of t, i.e.

\begin{align} \sqrt{S_t}(X_{S_t}/S_t-E(X_1))\xrightarrow{d}\mathcal{N}(0,\sigma^{2})\qquad \text{and mixing} \tag 2 \end{align}

When you look at the book dont be confused with their notation so i add the following identity $A_t/S_t=X_{S_t}/S_t$.

They use the following theorem cited from

Csörgö and Fischer (1973) "Some examples and results in the theory of mixing and random-sum central limit theorem"

Let $Y_n$ be mixing with limiting distribution $F$ and suppose, that $Y_n$ statisfies Anscombes condition (see below.) Let $K_{n}$ be a positive random variable with $K_n\rightarrow \infty$ a.s. and $f_n$ a monotonically increasing function to $\infty$ such that $K_n/f_n\rightarrow \nu$ a.s., where $\nu$ is a positive random variable. Then $Y_{K_n}$ is mixing with limit distribution $F$.

Anscombe Condition Given $\epsilon>0,\eta>0$, there exists a positive number c and an integer $n_{0}$ such that for all $n\geq n_{0}$

\begin{align} P\left(\max_{|m-n|<nc} |Y_{n}-Y_{m}|>\epsilon\right)<\eta \end{align}

Mixing convergence Given $Y_n\xrightarrow{d} Y$ in distribution. Both defined on some prob. space $(\Omega,\mathcal{A},P)$. We call the convergence mixing if for all $\mathcal{A}$-mb. sets $A$ hold that, \begin{align} \lim_{n\rightarrow \infty}P(Y_n\in B,A)=P(Y\in B)P(A)\,. \end{align} where $B$ are the countable dense sets of continuity of the according distribution function. So mixing convergence implies weak convergence.

MY Question Why is it legit to use the stated discrite time theorem for a continuous time setup considered now in the case of Sørensen? So far I haven't even found a definition of the Anscombe condition in continuous time. So I wonder if one does the proof in Sørensens setup generally in discrete time and concludes by the mixing convergence and concludes somehow that this holds for continuous time too.

Edit: I got the following idea: $$X_n=\sum_{i=1}^n X_1$$ is considered (due to the Lévy property) as a partial sum of $n$ i.i.d. variables. Then by CLT we have $$\sqrt{n}\left(\frac{X_n}{n}-E[X_1]\right) \rightarrow \mathcal{N}\left(0,\sigma^{2}\right) \text{ as } n\rightarrow \infty$$ Let $S_{1},S_{2},\ldots$ be the sequence of Values of $S_{t}$ at the integer discrete times. Then we can apply the CLT in the sense of Csörgö.

$S_n:=\lfloor S_t \rfloor$ \begin{align} \frac{X_{S_t}}{S_{t}}=\frac{S_n}{S_t}\left(\frac{X_{S_n}}{S_n}+\frac{X_{S_t}-X_{S_n}}{S_n}\right) \tag 3 \end{align} So considering the CLT we end up here \begin{align} \sqrt{S_t} \left(\frac{S_n}{S_t}\left(\frac{X_{S_n}}{S_n}+\frac{X_{S_t}-X_{S_n}}{S_n}\right)-E[X_1]\right) \end{align} This is exactly (2). With $t\rightarrow \infty$ so $S_{t}\rightarrow \infty$ and $S_n:=\lfloor S_t \rfloor\rightarrow \infty$ we can conclude the result from the discrete case, if we show that the second term in (3) tends to zero.

I appreciate any help, best regards.

Let $$Y_t := \sqrt{t}(X_t/t - \mathbb{E}[X_1])= \frac{X_{t} - t\mathbb{E}[X_1]}{\sqrt{t}}.$$

By applying the discrete versions of above theorems we have $Y_{\lfloor S_t\rfloor} \Rightarrow Y$ for $t \rightarrow \infty$ with $Y \sim \mathcal{N}(0,\sigma^2)$ and the convergence is mixing.

We rewrite

$$Y_{S_t} = \frac{X_{S_t} - S_t\mathbb{E}[X_1]}{\sqrt{S_t}} =$$ $$= \underbrace{\frac{\sqrt{\lfloor S_t \rfloor}}{\sqrt{S_t}}}_{\rightarrow 1 \text{ a.s.}}\bigg( \underbrace{\frac{X_{\lfloor S_t \rfloor} - \lfloor S_t \rfloor \mathbb{E}[X_1]}{\sqrt{\lfloor S_t \rfloor}}}_{= Y_{\lfloor S_t \rfloor} \Rightarrow Y} + \underbrace{\frac{\sqrt{f_t}}{\sqrt{\lfloor S_t\rfloor}}}_{\rightarrow \nu \text{ a.s.}} \frac{\overbrace{X_{S_t} - X_{\lfloor S_t \rfloor}}^{\stackrel{d}{=}X_{(S_t - \lfloor S_t \rfloor)}} - (S_t - \lfloor S_t \rfloor) \mathbb{E}[X_1]}{\sqrt{f_t}}\bigg)$$

It remains to show that the last term converges to 0. Note that $0 \leq S_t - \lfloor S_t \rfloor < 1$ a.s. and that we can assume that $X_{(S_t - \lfloor S_t \rfloor)} - (S_t - \lfloor S_t \rfloor) \mathbb{E}[X_1]$ is a martingale (see also text before Thm. 5.2.2 in M. Sørensen). Let $\varepsilon > 0$, then by using Doob's inequality we get

$$\mathbb{P}\left( \left\vert\frac{X_{(S_t - \lfloor S_t \rfloor)} - (S_t - \lfloor S_t \rfloor) \mathbb{E}[X_1]}{\sqrt{f_t}}\right\vert > \varepsilon\right) \leq \mathbb{P}\left(\sup_{0\leq s < 1} \vert X_s - s\mathbb{E}[X_1] \vert > \varepsilon\sqrt{f_t}\right)\leq \frac{\mathbb{E}[\vert X_1 - \mathbb{E}[X_1]\vert]}{\varepsilon\sqrt{f_t}} \rightarrow 0.$$

Also, I think that the proofs by Csörgö and Fischler should also work for continuously indexed processes, but above should give a direct answer to your question.