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
- Do Billingsley's Probability and Measures and his Convergence of Measures define the process $Y_n$ differently?
- 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?)
- How are the two versions of functional central limit theorems from Billingsley and the version from Kallenberg related and different?
- 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:
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 $\epsilon$.
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.
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.
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!
