# Different versions of functional central limit theorem (aka Donsker theorem)?

I have seen several versions of functional central limit theorem (see the end of this post). I am confused, and hope someone could help to clarify their relations and differences. For example, I am wondering

1. Do Billingsley's Probability and Measures and his Convergence of Measures define the process $Y_n$ differently?
2. Are the conclusion in Billingsley's Convergence of Measures different from the conclusion in his Probaility and Measures, in that the latter says an equivalent process to $Y_n$ but defined on another probability space converges to a Wiener process weakly, while the former says $Y_n$ converges a Wiener process in probability? (Note that convergence in probability implies weak convergence?)
3. How are the two versions of functional central limit theorems from Billingsley and the version from Kallenberg related and different?
4. How is Wikipedia's Donsker theorem related to the two versions of functional central limit theorems from Billingsley and the version from Kallenberg? I am not able to see if $G_n$ can be written as $Y_n$, and also $G_n$ converges in distribution to a Gaussian process which might not be a Wiener process?

PS: Three versions of functional central limit theorems:

1. In Billingsley's Probability and Measures:

Theorem 37.8. Suppose that $X_1,X_2,·-·$ are independent, identically distributed random variables with mean $0$, variance $σ^2$, and finite fourth moments, and define $Y_n(t), 0\leq t \leq 1$ by $$Y_n(t, \omega) = \frac{1}{\sigma\sqrt{n}} S_k(\omega), \text{ if }\frac{k-1}{n} < t \leq \frac{k}{n}.$$ There exist (on another probability space), for each n, processes $[Z_n(t): 0 \leq t \leq 1]$ and $[W_n(t): 0 \leq t \leq 1]$ such that the first has the same finite-dimensional distributions as $[Y_n(t): 0 \leq t \leq 1]$, the second is a Brownian motion, and $P[\sup_{t\leq 1} |Z_n(i) - W_n(t)| \geq \epsilon] \to 0$ for positive $\epsilon$.

2. In Billingsley's Convergence of Measures

Theorem 8.2. If $X_1, X_2,...$ are independent and identically distributed with mean 0 and variance $\sigma^2$, and if $Y_n$ is the random function. defined by $$Y_n(t, \omega) = \frac{1}{\sigma \sqrt{n}} S_{\lfloor nt \rfloor}(\omega) +(nt - \lfloor nt \rfloor) \frac{1}{\sigma \sqrt{n}} X_{\lfloor nt \rfloor + 1}(\omega), \quad 0\leq t \leq 1$$ then $Y_n$ converges to the Wiener process $W$ weakly.

3. In Kallenberg's Foundations of Probability Theory

Theorem 14.9 (functional central limit theorem, Donsker) Let $X_1, X_2, \dots$ be i. i. d. random variables with mean 0 and variance 1, and define $$Y_n(t) = \frac{1}{\sqrt{n}} \sum_{k \leq nt}X_k, t \in [0,1], n \in \mathbb N$$ Consider a Brownian motion $B$ on $[0, 1]$, and let $f : D[0, 1] \to \mathbb R$ be measurable and a.s. continuous at $B$. Then $f(Y_n) \to f(B)$ in distribution.

4. From Wikipedia

Donsker's theorem identifies a certain stochastic process as a limit of empirical processes. It is sometimes called the functional central limit theorem.

A centered and scaled version of empirical distribution function Fn defines an empirical process $$G_n(x)= \sqrt n ( F_n(x) - F(x) ) \,$$ indexed by $x ∈ \mathbb R$.

Theorem (Donsker, Skorokhod, Kolmogorov) The sequence of $G_n(x)$, as random elements of the Skorokhod space $\mathcal{D}(-\infty,\infty)$, converges in distribution to a Gaussian process $G$ with zero mean and covariance given by $$\operatorname{cov}[G(s), G(t)] = E[G(s) G(t)] = \min\{F(s), F(t)\} - F(s)F(t). \,$$ The process $G(x)$ can be written as $B(F(x))$ where $B$ is a standard Brownian bridge on the unit interval.

Thanks and regards!

• hello, you seem to be beginer in probability just like me...i thought you were tim austin (terence tao student)....neways nice to see like minded people over here – user24367 Oct 27 '13 at 11:25
• @user17523: Thanks for your comment. I am just an admirer of Terry Tao, and don't have a chance to pursue a graduate study in probability right now. That is my biggest pain. – Tim Oct 28 '13 at 18:07
• so you completed undergraduate in maths or engineering? – user24367 Oct 28 '13 at 21:32
• undergraduate and master in Engineering. Are you a graduate student in stochastic processes? Which university? – Tim Oct 28 '13 at 21:48
• I did undergraduate from India in ECE….then masters in ECE from Notre Dame, completed it in 2011 (nothing stochastic till this time, mostly Information Theory and Network Information Theory, though I worked in control, and simple stochastic stuff like expectations sufficed to do the work...)…..moved to Texas A&M two years ago (got RA from very reputed prof who recently moved A&M after 30 years in top unit in IL, since then studied Real Analysis (graduate level directly, since had no undergraduate level real analysis)..now working in queueing theory and reading advanced probability on own – user24367 Oct 29 '13 at 0:01