# Functional Central Limit Theorem

My question is related to the definitions of Functional Central Limit Theorem mentioned on Wikipedia

Version 1: If $X_1,X_2,X_3,\dots,X_n$ are iid random variables with mean 0 and variance 1, then $S_n(t) = \frac{\sum_1^{\lfloor tn\rfloor} X_i}{\sqrt n}$ converges in distribution to a standard brownian motion.

Version 2: Let $F_n(x)$ be the empirical distribution function of $F(X)$. Define $G_n(X) = \sqrt{n}(F_n(x)- F(x))$. Then $G_n(X)$ converges in distribution to a standard brownian bridge.

• I would start by checking what you've written, since it looks to me like neither version is quite the same as what's written on the wikipedia page. In V1, does it say that $S_{n}$ converges in dist. to a st. Br. mot? In V2, the way you've defined $G_{n}(X)$ doesn't make sense as written, since the left side is independent of $x$ but the right hand side depends on $x.$ – RideTheWavelet Jan 10 '18 at 18:29
• And in Version 1, you misstated the upper summation bound: it is $\lfloor tn\rfloor$ not $n$. – kimchi lover Jan 10 '18 at 18:37