0.1 Convergence in Law
Convergence in Law, also known as convergence in distribution or weak convergence is the fourth and weakest type of convergence of random variables.
A sequence of random variables \(\{X_n\}_{n\in\mathbb{N}}\) on probability space \((\Omega, \mathcal{F}, \mathbb{P})\) is said to converge in law (or converge in distribution) to \(X\) as \(n\to \infty\), denoted \(X_{n}\stackrel{\mathcal{D}}{\to}X\) if \[ \lim_{ n \to \infty } F_{X_{n}}(x)=F_{X}(x), \] that is, their sequence of [[distribution-function|distribution functions]] converge.
We note the following key points and observations about convergence in law:
Unlike other types of convergence (a.s., norm, probability) convergence in law tells us nothing about the behavior of the random variables themselves, only their distribution. See [[files-notes-pstat213bc-lecture-notes-raya-pdf|PSTAT213BC Lecture Notes (Raya), page 32]] for an explanatory example.
Convergence in law means that \(F_{n}(x) \to F(x)\) for all \(x\) up to the points of discontinuity of \(F\).
Limits in distribution are unique, that is \(F_{n}(x) \to F(x)\) and \(F_{n}(x) \to G(x)\) then \(F=G\). ### Convergence in Probability \(\implies\) Convergence in Law
If \(X_{n}\stackrel{{\mathbb{P}}}{\to}X\) as \(n \to \infty\), then \(X_{n}\stackrel{\mathcal{D}}{\to}X\) as \(n\to \infty\).
\begin{proof} For every \(x \in\mathbb{R}\), \(\epsilon>0\) we have that \[
\mathbb{P}(X\leq x-\epsilon)\leq \liminf_{n \to \infty } \mathbb{P}(X_{n}\leq x)\leq \limsup_{ n \to \infty }\mathbb{P}(X_{n}\leq x)\leq \mathbb{P}(X\leq x+\epsilon).
\] If \(x\) is a continuity point of \(X\), then as \(\epsilon\downarrow 0\), the left and right sides converge and hence \[
\lim_{ n \to \infty } \mathbb{P}(X_{n}\leq x)=P(X\leq x),
\] giving the result.\end{proof} ### Skorohod's Representation Theorem
Skorohod’s Representation Theorem is a result showing that a convergent in law sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution / law of a pointwise convergent sequence of random variables defined on a common probability space.
Suppose that \(X_{n}\stackrel{\mathcal{D}}{\to}X\) as \(n \to \infty\) with \(F_{n}(x):=\mathbb{P}(X_{n} \leq x)\) and \(F(x):=\mathbb{P}(X\leq x)\) for \(x \in \mathbb{R}\). Then, there exists a probability space \((\Omega', \mathcal{F}',\mathbb{P}')\) and random variables \(\{Y_{n}, n\geq 1\}\) and \(Y\) such that \[ Y_{n}\stackrel{\mathcal{D}}{=}X_{n},\quad Y\stackrel{\mathcal{D}}{=X}, \] for every \(n\) and \(Y_{n}\stackrel{a.s.}{\to}Y\) as \(n \to \infty\).
0.2 Key Results
With convergence of distribution functions we consider the following questions:
- Does a sequence of d.f.s \(\{ F_{n} \}\) necessarily converge?
- In general no, Helly's Selection Theorem provides a partial answer.
- If a sequence of d.f.s \(\{ F_{n} \}\) converges to a function \(F\): \(F_{n}(x) \to F(x)\) is the limit \(F\) necessarily a distribution function?
- In general, no, the space of distribution functions is not compact.
- In general, no, the space of distribution functions is not compact.
- When is the limit \(F\) (of a sequence of d.f.s \(\{ F_{n} \}\)) is a proper d.f.?
- When \(\{ F_{n} \}\) is tight.
Tightness ##### Prohorov’s Theorem