8
$\begingroup$

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.

$\endgroup$

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.