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Denote $S_n := \sum_{k=1}^n X_i$, where the random variables $X_i$ are independent and identically distributed. Suppose $\mathbb{E}[X_i] = \mu$ and $\mathbb{V}[X_i] = \sigma^2$ are finite. Let $\lfloor x \rfloor$ be the integer part of $x \geq 0$. Define $\mathcal{D}$ as the space of real càdlàg functions on $[0, 1]$. Denoting $B(t)$ standard Brownian motion (or the Wiener process), it should follow from Donsker's invariance principle that \begin{equation} \lim_{n \to \infty} \left\{ \frac{S_{\lfloor n t \rfloor}}{\sqrt{n}} \right\} = t \mu + \sigma B(t) \sim \text{Normal}[t \mu, t \sigma^2] \quad\forall t \in [0, 1] \end{equation} in the space $\mathcal{D}[0, 1]$ with suitable metric.

Suppose I multiply $t$ by some non-negative random variable $T$ prior to scaling. Taking the limit as above, under which circumstances, if any, do I get as result $T t \mu + \sigma B(Tt)$?

I'm not a mathematician, so this seems too naive to be true. Is there a way to modify the argument to characterize the universe of subordinate Brownian motions with drift? I am particularly interested in the Variance-Gamma process, where $T \sim \text{Gamma}(1/\nu, \nu)$ so $\mathbb{E}[T] = 1$ and $\mathbb{V}[T] = \nu$.

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  • $\begingroup$ As a follow-up question, why is the whole apparatus of triangular arrays necessary to treat limits with random sample sizes/random time changes? I suppose one cannot "exchange limits" as I do above in some sense (i.e., I first let the partial-sum process converge to Brownian motion and afterwards randomize time; I suppose that is illegal in some rigorous sense, or does not exhaustively prove convergence towards the process for all $t$). But it would be important for me to understand where exactly the error lies. $\endgroup$
    – bodhi
    Apr 21, 2022 at 1:18

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