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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.

Can you please help me understand how both these versions are equivalent?

Thank you in advance

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  • $\begingroup$ 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 $x.$ $\endgroup$ – RideTheWavelet Jan 10 '18 at 18:29
  • $\begingroup$ And in Version 1, you misstated the upper summation bound: it is $\lfloor tn\rfloor$ not $n$. $\endgroup$ – kimchi lover Jan 10 '18 at 18:37
  • $\begingroup$ @kimchilover: Edited! Thank you! $\endgroup$ – kasa Jan 10 '18 at 18:56
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There's obviously two separate (but related) results here, and the Wikipedia article is not very clear. Better is Billingsley's Convergence of Probability Measures, which discusses them in separate chapters. The first it what is usually called the "functional central limit theorem" or "the invariance principle". The cumulative sum curves converge in distribution to Brownian motion paths. The second is about empirical cumulative distribution functions, or empirical measures (which are the subject of the Glivenko-Cantelli theorem and the Kolmogoroff-Smirnov theorem): the rescaled indefinite integrals of the empirical histograms converge in distribution to the "Brownian bridge" or "pinned Brownian motion". Both are known as "Donsker's theorem".

Because M. D. Donsker proved both. His first result was published in "An invariance principle for certain probability limit theorems." Mem. Amer. Math. Soc., No. 6 (1951). His second one was published in "Justification and extension of Doob's heuristic approach to the Komogorov-Smirnov theorems", Ann. Math. Statistics 23, (1952), pp. 277–281.

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