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.