Limit of Cumulative Distribution Function at Positive Infinity

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Theorem

Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$.

Let $F_X$ be the cumulative distribution function.


Then:

$\ds \lim_{x \mathop \to \infty} \map {F_X} x = 1$

where $\ds \lim_{x \mathop \to \infty} \map {F_X} x$ denotes the limit at $+\infty$ of $F_X$.


Proof

From Sequential Characterization of Limit at Positive Infinity of Real Function: Corollary, we aim to show that:

for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map {F_X} {x_n} \to 1$

at which point we will obtain:

$\ds \lim_{x \mathop \to \infty} \map {F_X} x = 1$

Since $\sequence {x_n}_{n \mathop \in \N}$ is increasing:

the sequence $\sequence {\hointl {-\infty} {x_n} }_{n \mathop \in \N}$ is increasing.

We show that:

$\ds \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \R$

Lemma

Let $\sequence {x_n}_{n \mathop \in \N}$ be an increasing with $x_n \to \infty$.

Then:

$\ds \bigcup_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \R$

$\Box$


Let $P_X$ be the probability distribution of $X$.

So, from Measure of Limit of Increasing Sequence of Measurable Sets, we have:

$\ds \map {P_X} \R = \lim_{n \mathop \to \infty} \map {P_X} {\hointl {-\infty} {x_n} }$

So, we obtain:

\(\ds 1\) \(=\) \(\ds \map {P_X} \R\) Probability Distribution is Probability Measure, Definition of Probability Measure
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {P_X} {\hointl {-\infty} {x_n} }\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \map \Pr {X \le x_n}\) Definition of Probability Distribution
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \map {F_X} {x_n}\) Definition of Cumulative Distribution Function

Since $\sequence {x_n}_{n \mathop \in \N}$ was arbitrary, we have:

for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map {F_X} {x_n} \to 1$.

So, from Sequential Characterization of Limit at Positive Infinity of Real Function: Corollary, we have:

$\ds \lim_{x \mathop \to \infty} \map {F_X} x = 1$

$\blacksquare$


Sources

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