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 $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.