# Donsker's Theorem for triangular arrays

I should mention that I already posed this question on Math Stack Exchange, but didn't receive much feedback.

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $(\frac{X_i}{n^{\alpha}})_{i=1}^n$? More precisely, does

$$n^{2\alpha}\left(\frac{\sum_{i=1}^n \mathbf 1_{\{X_i\leq t n^{-\alpha}\}}}{n} - F_X(tn^{-\alpha})\right)\stackrel{\mathrm d}{\rightarrow} B(t),$$

where $B(\cdot)$ is a Brownian motion and $F_X$ is the distribution function of $X_1$, hold? I should mention that the $n^{2\alpha}$ is only an (informed) guess. In the usual Donsker's Theorem the limiting process is $B_0(F(t))$, where $B_0$ is a Brownian Bridge, but in this case the limiting process would need to be pinned down to zero 'at infinity', and thus my guess that it's actually a Brownian Motion.

Is this common knowledge? I have been digging through the literature and it does not seem to be proved anywhere.

• Maybe the paper sciencedirect.com/science/article/pii/S0304414997001026 by Arcones can be useful. Nov 21, 2014 at 17:47
• @DavideGiraudo Thank you for your suggestion. It was helpful. It does seem, however, that the results there are more suited for compactly supported processes and not processes on $[0,+\infty)$ (he uses finite partitions of the parameter space $T$). Nov 26, 2014 at 10:58
• If you consider process indexed by $[0,\infty)$, you may want to give a functional space on which you want the weak convergence. Nov 26, 2014 at 11:33
• My guess would be that it should be provable for the space of cadlag functions on $[0,\infty)$ with some Skorohod topology (possibly $J$). Nov 26, 2014 at 16:43

I guess you assume the $X_i$'s to take values in $[0,\infty)$. As it seems you are essentially rescaling in time as well, I would rather expect a convergence to a Poisson process.

Take for example $\alpha=1$. Then it is known that

$\sum_{i=1}^n \mathbf{1}_{\{X_i \leq tn^{-1}\}} \stackrel{d}{\longrightarrow} N(t),$

where $(N(t))_{t\geq 0}$ is a Poisson process with intensity function $f_X(0)t$ and $f_X$ is the density function associated with $F_X$ (see for example Thm. 4.41 in [1]; for more details and stronger types of convergence see e.g. this paper).

Let's assume your your type of Donsker's theorem was correct. If we take $X$ to be exponentially distributed, say $F_X(x) = 1 - \exp(-x)$, then we can show via power series expansion $n^2F_X(t/n) = \mathcal{O}(n)$. But this would yield for $n \rightarrow \infty$

$$\label{eq:2} n\Bigg(\underbrace{\sum_{i=1}^n \mathbf{1}_{\{X_i \leq tn^{-1}\}}}_{\rightarrow N(t)} + \,\mathcal{O}(1)\Bigg) \rightarrow +\infty.$$

References

[1] Jacod, J. and A. N. Shiryaev (2003). Limit theorems for stochastic processes (Second ed.)