# From Donsker's theorem to subordinate Brownian motion

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 $$$$\lim_{n \to \infty} \left\{ \frac{S_{\lfloor n t \rfloor}}{\sqrt{n}} \right\} = t \mu + \sigma B(t) \sim \text{Normal}[t \mu, t \sigma^2] \quad\forall t \in [0, 1]$$$$ in the space $$\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 $$\mathbb{V}[T] = \nu$$.

• 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. Apr 21, 2022 at 1:18