Tight Probability Processes 📂Probability Theory

Tight Probability Processes

Definition

Let us say in a probability space $( \Omega , \mathcal{F} , P)$ , a stochastic process $\left\{ X_n \right\}_{n \in \mathbb{N}}$ is defined. If for every $\varepsilon > 0$ , there exists a compact set $K \subset \Omega$ such that $\displaystyle \inf_{n \in \mathbb{N}} P\left( X_{n} \in K \right) > 1 - \varepsilon$ is satisfied, then $\left\{ X_{n} \right\}$ is said to be tight.

Explanation

In mathematical statistics, it corresponds to the concept of boundedness in probability. Tightness is important in relation to convergence in distribution, possessing several crucial properties as follows.

Fundamental Properties

Let $X$ , $\left\{ X_n \right\}_{n \in \mathbb{N}}$ be a probabilistic element and stochastic process defined in a metric space $(S, d)$ and let $\mathscr{H}: = C(S, \mathbb{R})$ .

[1]: If $\left\{ X_{n} \right\}$ is tight, it is precompact.
[2]: If $\left\{ X_{n} \right\}$ is tight, then for all $h \in \mathscr{H}$ , if $h(X_{n}) \overset{D}{\to} h(X)$ , then $X_{n} \overset{D}{\to} X$

Assuming $X$ , $\left\{ X_n \right\}_{n \in \mathbb{N}}$ are probabilistic element and stochastic process defined in $C[0,1]$ respectively.

[3]: Supposing that $X$ is a probabilistic element in $S = C[0,1]$ . If at all finite subsets $A$ of points $a$ in $[0,1]$ , $X_{n}(a) \overset{W}{\to} X(a)$ and $\left\{ X_{n} \right\}$ is tight, then $X_{n} \overset{D}{\to} X$
[4]: That $\left\{ X_{n} \right\}$ is tight implies (i) for all $\varepsilon > 0$ , $\lim_{\delta \to 0} \limsup_{n \to \infty} P \left( \sup_{|s-t| < \delta} \left| X_{n}(s) - X_{n}(t) \right| \ge \varepsilon \right) = 0$ and (ii) $\left\{ X_{n} (0) \right\}$ being tight is equivalent.

$C[0,1]$ represents the space of continuous functions with domain $[0,1]$ and codomain $\mathbb{R}$ .
$C(S,\mathbb{R})$ is the space of continuous functions with domain $S$ and codomain $\mathbb{R}$ .