Random Variables

Inverse Images

For any function $X : \Omega \to \Omega'$ and $A' \subseteq \Omega'$,

$$X^{-1}(A') = \{\omega \in \Omega : X(\omega) \in A'\} = [X \in A']$$

Inverse images preserve set operations,

$$X^{-1}(\emptyset) = \emptyset, \quad X^{-1}(\Omega') = \Omega, \quad X^{-1}((A')^c) = (X^{-1}(A'))^c, \quad X^{-1}\Big(\bigcup_{j \in T} A'_j\Big) = \bigcup_{j \in T} X^{-1}(A'_j)$$

and the same for intersections. Hence if $\mathcal{B}'$ is a $\sigma$-field on $\Omega'$, $X^{-1}(\mathcal{B}')$ is a $\sigma$-field on $\Omega$, and for $\mathcal{C}' \subseteq 2^{\Omega'}$,

$$X^{-1}(\sigma(\mathcal{C}')) = \sigma(X^{-1}(\mathcal{C}'))$$

Measurable Functions

A measurable space is a pair $(\Omega, \mathcal{B})$ where $\mathcal{B}$ is a $\sigma$-algebra of subsets of $\Omega$.

For measurable spaces $(\Omega, \mathcal{B})$ and $(\Omega', \mathcal{B}')$, $X : \Omega \to \Omega'$ is measurable if $X^{-1}(B') \in \mathcal{B}$ for all $B' \in \mathcal{B}'$. Equivalently $X^{-1}(\mathcal{B}') \subseteq \mathcal{B}$, written $X \in \mathcal{B}/\mathcal{B}'$. If $(\Omega', \mathcal{B}') = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ then $X$ is a random variable (RV).

It is enough to check generators. If $\mathcal{B}' = \sigma(\mathcal{C}')$, then $X$ is measurable iff $X^{-1}(C') \in \mathcal{B}$ for all $C' \in \mathcal{C}'$. So $X : \Omega \to \mathbb{R}$ is an RV iff

$$[X \le t] = X^{-1}((-\infty, t]) \in \mathcal{B} \quad \forall t \in \mathbb{R}$$

Distribution

Given $(\Omega, \mathcal{B}, P)$ and a measurable $X : (\Omega, \mathcal{B}) \to (\Omega', \mathcal{B}')$, the distribution of $X$ is $Q = P \circ X^{-1}$,

$$Q(B') = P(X^{-1}(B')) = P([X \in B']), \quad B' \in \mathcal{B}'$$

Generalized Inverse

For a df $F$ and $0 < u < 1$,

$$F^{\leftarrow}(u) = \inf\{s \in \mathbb{R} : F(s) \ge u\}$$

$F^{\leftarrow}$ is non-decreasing and left-continuous, and for $u \in (0,1)$, $t \in \mathbb{R}$,

$$F^{\leftarrow}(u) \le t \iff u \le F(t)$$

Metric Spaces

$d : S \times S \to [0, \infty)$ is a metric if for $x, y, z \in S$

1. $d(x,y) = 0$ iff $x = y$,
2. $d(x,y) = d(y,x)$,
3. $d(x,z) \le d(x,y) + d(y,z)$.

For $x \in S$ and $\varepsilon > 0$, $B_\varepsilon(x) = \{y \in S : d(x,y) < \varepsilon\}$. A set $A$ is open if for all $x \in A$ there is $\varepsilon > 0$ such that $B_\varepsilon(x) \subseteq A$. Let $\mathcal{O}$ be the class of all open sets.

The Borel $\sigma$-algebra of $(S, d)$ is $\mathcal{B}(S) = \sigma(\mathcal{O})$, and a measurable $X : (\Omega, \mathcal{B}) \to (S, \mathcal{B}(S))$ is an $S$-valued random element.

Building Measurable Functions

• Continuous functions are Borel measurable For metric spaces $S_1, S_2$ with Borel $\sigma$-algebras, a continuous $X : S_1 \to S_2$ is measurable.
• Composition If $X : (\Omega, \mathcal{B}) \to (S, \mathcal{S})$ and $f : (S, \mathcal{S}) \to (T, \mathcal{T})$ are measurable, so is $f \circ X$.
• $X = (X_1, \ldots, X_k)$ is a random vector iff each $X_i$ is an RV. So $X + Y$, $XY$, $\max(X,Y)$ are RVs.
• For real valued RVs, $\sup_{n \ge 1} X_n$, $\inf_{n \ge 1} X_n$, $\liminf_{n \to \infty} X_n$ and $\limsup_{n \to \infty} X_n$ are RVs. If $X_n(\omega)$ converges to a finite limit for all $\omega$, $X = \lim_{n \to \infty} X_n$ is an RV. Also $\{\omega : \lim_{n \to \infty} X_n(\omega) \text{ exists}\} \in \mathcal{B}$.

A simple function is a measurable real valued function with finite range.

$\sigma(X)$

For measurable $X : (\Omega, \mathcal{B}) \to (\Omega', \mathcal{B}')$,

$$\sigma(X) = X^{-1}(\mathcal{B}') = \{[X \in B'] : B' \in \mathcal{B}'\}$$

For a real RV the target $\sigma$-algebra is $\mathcal{B}(\mathbb{R})$.

A class $\mathcal{F} \subseteq \mathcal{B}$ is a sub $\sigma$-algebra if it is itself a $\sigma$-algebra on $\Omega$. A random element $X$ is $\mathcal{F}$-measurable if $\sigma(X) \subseteq \mathcal{F}$, i.e. $X^{-1}(B') \in \mathcal{F}$ for all $B' \in \mathcal{B}'$, shorthand $X \in \mathcal{F}$.

For a family $\{X_t : t \in T\}$ on $(\Omega, \mathcal{B})$,

$$\sigma(X_t : t \in T) := \bigvee_{t \in T} \sigma(X_t) = \sigma\Big(\bigcup_{t \in T} \sigma(X_t)\Big)$$

the smallest $\sigma$-algebra containing all $\sigma(X_t)$.

Example: for $A \in \mathcal{B}$ and $X = \mathbf{1}_A$, since $X^{-1}(\{1\}) = A$ and $X^{-1}(\{0\}) = A^c$,

$$\sigma(X) = \{\emptyset, \Omega, A, A^c\}$$

so observing $X$ tells whether $A$ occurred.

Generators for $\sigma(X)$

If $X : (\Omega, \mathcal{B}) \to (\Omega', \mathcal{B}')$ is measurable and $\mathcal{B}' = \sigma(\mathcal{C}')$, then

$$\sigma(X) = \sigma(\{[X \in C'] : C' \in \mathcal{C}'\})$$

If $X$ is real valued, $\sigma(X) = \sigma(\{[X \le x] : x \in \mathbb{R}\})$.