Properties of RVs

We explore key properties of random variables, focusing on the $\sigma$-fields they generate and how they behave under transformations and limits.

The σ-Field Generated by $X$

Theorem

Statement

Consider the class of sets:

$$\mathcal{L} = \{ \{X \in B\} : B \in \mathcal{B} \} = \{ X^{-1}(B) : B \in \mathcal{B} \}$$

where $\{X \in B\}$ is shorthand for $\{\omega \in \Omega : X(\omega) \in B\}$. $\mathcal{L}$ is a $\sigma$-field on $\Omega$. Furthermore, it is the smallest $\sigma$-field on $\Omega$ which makes $X$ measurable. We denote it by $\sigma(X)$.

Proof

1. Entire Space: $\Omega = \{ X \in \mathbb{R} \} \in \mathcal{L}$ since $\mathbb{R} \in \mathcal{B}$. (Preimage of the codomain is the domain).

2. Complements: If $A = \{ X \in B \}$ for some $B \in \mathcal{B}$, then:

   $$A^c = \{ X \notin B \} = \{ X \in B^c \}$$

   Since $\mathcal{B}$ is a $\sigma$-field, $B^c \in \mathcal{B}$, so $A^c \in \mathcal{L}$.

3. Countable Unions: If $A_1, A_2, \dots \in \mathcal{L}$ such that $A_i = \{ X \in B_i \}$ for $B_i \in \mathcal{B}$, then:

   $$\bigcup_n A_n = \bigcup_n \{ X \in B_n \} = \{ X \in \bigcup_n B_n \}$$

   Since $\bigcup B_n \in \mathcal{B}$, the union $\bigcup A_n \in \mathcal{L}$.

This concept generalizes to multiple random variables. $\sigma(X, Y)$ is the smallest $\sigma$-field making both $X$ and $Y$ measurable.

Transformations of Random Variables

A fundamental property is that functions of random variables are themselves random variables.

Theorem: Composite Mapping

Statement

If $X: (\Omega, \mathcal{F}) \to (S, \mathcal{A})$ and $f: (S, \mathcal{A}) \to (T, \mathcal{B})$ are measurable mappings, then the composition $f(X)$ is also measurable from $(\Omega, \mathcal{F})$ to $(T, \mathcal{B})$.

Proof

For any $B \in \mathcal{B}$, we want to show that $\{\omega : f(X(\omega)) \in B\} \in \mathcal{F}$. Observe:

$$\{ \omega : f(X(\omega)) \in B \} = \{ \omega : X(\omega) \in f^{-1}(B) \}$$

Since $f$ is measurable, let $A = f^{-1}(B) \in \mathcal{A}$. Then the set becomes $\{ \omega : X(\omega) \in A \}$. Since $X$ is measurable, this set is in $\mathcal{F}$.

Corollary

If $X_1, \dots, X_n$ are random variables, then:

• $-X$, $|X|$, $X^2$
• $X_1 + X_2$, $X_1 X_2$
• $e^X$, $\sin(X)$ are all random variables.

Limits of Random Variables

Theorem: Limits

Statement

If $X_1, X_2, \dots$ are random variables, then the following are also random variables:

• $\inf_n X_n$
• $\sup_n X_n$
• $\liminf_{n \to \infty} X_n$
• $\limsup_{n \to \infty} X_n$

Proof Idea

The key is that logical quantifiers correspond to set operations (unions and intersections).

1. Infimum: Observe that $\inf_n X_n < a$ if and only if there exists some $n$ such that $X_n < a$.

   $$\{\inf_n X_n < a\} = \bigcup_{n} \underbrace{\{ X_n < a \}}_{\in \mathcal{F}}$$

   Since $\mathcal{F}$ is a $\sigma$-field, the countable union is in $\mathcal{F}$. Thus, the infimum is measurable.

2. Supremum: Similarly, $\sup_n X_n > a$ if and only if there exists some $n$ such that $X_n > a$.

   $$\{\sup_n X_n > a\} = \bigcup_{n} \{ X_n > a \} \in \mathcal{F}$$

3. Liminf & Limsup: Recall that:

   $$\liminf_{n \to \infty} X_n = \sup_{n \ge 1} \left( \inf_{m \ge n} X_m \right)$$

   Let $Y_n = \inf_{m \ge n} X_m$. By the Infimum property, each $Y_n$ is a random variable. Then $\liminf X_n = \sup_n Y_n$. By the Supremum property, this supremum is a random variable.

   The same logic applies to $\limsup X_n$.

Convergence

When does the limit $\lim_{n \to \infty} X_n$ exist? It exists exactly when the limit inferior equals the limit superior. Consider the set where the limit exists:

$$\Omega_0 = \{ \omega : \lim_{n \to \infty} X_n(\omega) \text{ exists} \}$$

This event can be written as:

$$\Omega_0 = \{ \omega : \limsup_{n \to \infty} X_n(\omega) - \liminf_{n \to \infty} X_n(\omega) = 0 \}$$

Since $\limsup X_n$ and $\liminf X_n$ are random variables, their difference is a random variable (call it $Y$). Thus $\Omega_0 = \{ Y = 0 \}$ is a measurable event.

Almost Sure Convergence

If $\mathbb{P}(\Omega_0) = 1$, we say that the sequence $X_n$ converges almost surely.