SkorohodsRepresentationTheorem

Skorohod's theorem is a mathematical result in probability theory named after the Soviet mathematician Anatoliy Skorohod. The theorem provides a representation for the weak convergence of probability measures on a metric space. It states that if a sequence of random variables or probability measures converges weakly (in distribution) to a limiting random variable or probability measure, then there exists another sequence of random variables, defined on a common probability space, that converges almost surely to the limit.

The theorem has important implications in various areas of probability theory and stochastic processes, such as convergence of stochastic processes, weak convergence of measures, and functional analysis.

To pronounce "Skorohod," say it as "skuh-RAH-hod" (rhyming with "score ah rod"). The emphasis is on the second syllable, and the last syllable is pronounced like "hot."

More precisely

Skorohod's Representation Theorem can be stated more formally as follows:

Let (S, d) be a complete separable metric space, and let $P_n, n \geq 1$, and $P$ be Borel probability measures on $(S, B(S))$, where $B(S)$ denotes the Borel sigma-algebra on $S$. Suppose that $P_n$ converges weakly (in distribution) to $P$, i.e.,

$$\lim_{n \to \infty} \int_S f dP_n = \int_S f dP, \quad \text{for all bounded, continuous functions } f: S \to \mathbb{R}.$$

Then, there exists a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and random elements (random variables if $S = \mathbb{R}$) $X_n: \Omega \to S, n \geq 1$, and $X: \Omega \to S$, such that:

1. The distribution of $X_n$ is $P_n$, and the distribution of $X$ is $P$; that is, $\mathbb{P}(X_n \in B) = P_n(B)$ and $\mathbb{P}(X \in B) = P(B)$ for all $B \in B(S)$.
2. $X_n$ converges to $X$ almost surely (a.s.) with respect to $\mathbb{P}$; that is, $\mathbb{P}({\omega \in \Omega: X_n(\omega) \to X(\omega) \text{ as } n \to \infty}) = 1$.

In other words, Skorohod's Representation Theorem asserts that if a sequence of probability measures $P_n$ converges weakly to a probability measure $P$ on a complete separable metric space, then there exists a sequence of random elements $X_n$ and a random element $X$ defined on a common probability space, such that $X_n$ converges to $X$ almost surely, and their distributions match the given probability measures.