---- Here, the notation $\widetilde{X}$ denotes the [[distribution function]] of a [[random variable]] $X$. > [!theorem] Theorem. ([[characterization of cumulative distribution functions]]) > Suppose $H:\mathbb{R} \to [0,1]$ is a function. Then the following are equivalent: > > **$(\text{a})$.** There exists a [[probability|probability space]] $(\Omega, \mathcal{F}, \mathbb{P})$ and a [[random variable]] $X$ on $(\Omega, \mathcal{F})$ such that $H=\widetilde{X}$. > > **$(\text{b})$.** The function $H$ satisfies the following four properties: > 1. *([[monotonic map|increasing]])* $s<t$ $\implies$ $H(s) \leq H(t)$; > 2. *(lower boundary vanishes)* $\lim_{t \to -\infty} H(t)=0$; > 3. *(upper boundary saturates)* $\lim_{t \to \infty}H(t)=1$; > 4. *([[left-continuous, right-continuous|right-continuity]])* $\lim_{t \to s^+} H(t)=H(s)$ for every $s \in \mathbb{R}$ > > > [!proof]- Proof. ([[characterization of cumulative distribution functions]]) > **$(a)\implies (b)$.** Suppose $H=\widetilde{X}$ for some [[probability|probability space]] $(\Omega, \mathcal{F}, \mathbb{P})$ and some [[random variable]] $X$ on $(\Omega, \mathcal{F})$. We must show that $(1)-(4)$ holds. > > $(1):$ holds by [[measure|monotonicity of measure]]. > > $(2):$ Let $(t_{n}) \to -\infty$ be a [[sequence]] in $\mathbb{R}$. By the sequential characterization of [[function limit|function limits]] $\lim_{t \to -\infty}H(t)=\lim_{n \to \infty}t_{n}$. So $\begin{align} > \lim_{t \to -\infty}H(t)&= \lim_{n \to \infty}H(t_{n}) \\ > &= \lim_{n \to \infty} \mathbb{P}(X \leq t_{n}) \\ > &= \mathbb{P}(\bigcap_{n=1}^{\infty} \{ X \leq t_{n} \}) \\ > &= \mathbb{P}(\emptyset) \\ > &= 0, > \end{align}$ > where we have used [[measure|continuity from above]]. > > $(3):$ Let $(t_{n}) \to \infty$ be a sequence in $\mathbb{R}$. We have $\begin{align} > \lim_{t \to \infty} H(t) &= \lim_{n \to \infty} \mathbb{P}(X \leq t_{n}) \\ > &= \mathbb{P}(\bigcup_{n=1}^{\infty} \{ X \leq t_{n} \}) \\ > &= \mathbb{P}(\Omega) \\ > &= 1, > \end{align}$ > where we have used [[measure|continuity from below]]. > > $(4):$ Let $(t_{n})$ be a [[sequence]] in $\mathbb{R}-\{ s \}$ with $t_{n} \to s ^{+}$. By the sequential characterization of right-limits, $\lim_{t \to s ^{+}}=H(t)=\lim_{n \to \infty} H(t_{n})$. We further assume $(t_{n})$ [[monotonic map|monotone decreasing]], implying $\{ x \leq t_{n} \} \supset \{ x \leq t_{n+1} \}$ for all $n \in \mathbb{N}$. Noting that $\bigcap_{n=1}^{\infty} \{ X \leq t_{n} \}= \{ X \leq s \},$ > we may now apply [[measure|continuity from above]] to obtain $\begin{align} > \lim_{t \to s^+} H(t) &= \lim_{n \to \infty} \mathbb{P}\{ X \leq t_{n} \} \\ > &= \mathbb{P}(\bigcap_{n=1}^{\infty} \{ X \leq t_{n} \}) \\ > &= \mathbb{P}(X \leq s) = H(s). > \end{align}$ > > **$(b) \implies (a)$.** Suppose $H:\mathbb{R} \to [0,1]$ is a function satisfying $(1)$-$(4)$ . With $\Omega=(0,1)$, $\mathcal{F}$ the collection of [[Borel set|Borel subsets]] of $(0,1)$, and $\mathbb{P}$ the [[Lebesgue measure]] on $(0,1)$, define a [[random variable]] $X:\Omega \to \mathbb{R}$ as[^1] $X(\omega):= \sup \{ t \in \mathbb{R}: H(t)< \omega \} \ (*)$Let $s \in \mathbb{R}$. We claim $H(s)=\widetilde{X}(s)$. To do this, we will show $\{ X \leq s \}=(0, H(s)]$. > > **$\supset.$** If $\omega \in (0, H(s)]$, then $X(\omega) \leq X(H(s))=\sup_{H(t)<H(s)} t \leq s$where the first inequality used that $X$ is increasing and the second used that $H$ is increasing. > > **$\subset.$** If $\omega \in (0,1)$ is such that $X(\omega) \leq s$, then $H(t) \geq \omega$ for all $t>s$ (by the contrapositive of $(*)$). Thus $H(s)=\lim_{t \to s^{+}} H(t) \geq \omega,$ > where the equality is right-continuity of $H$. Rewriting the inequality above, we have $\omega \in (0, H(s)]$, as required. [^1]: Clearly $X$ is an increasing function and [[σ-algebra generated by a set collection|thus]] is [[measurable function|measurable]] (hence indeed is a [[random variable]]). ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```