# Prove that the tail of this distribution goes to zero

• I
• CGandC
In summary, the theorem states that the limit of the probability of a random variable being greater than or equal to a certain value approaches zero as that value approaches infinity. The proof uses Heine's definition of limit and a monotonically increasing sequence to show this. The equation used in the proof is derived from the law of continuity of probability.f

#### CGandC

Theorem: Let ## X ## be a random variable. Then ## \lim_{s \to \infty} P( |X| \geq s ) =0 ##

Proof from teacher assistant's notes: We'll show first that ## \lim_{s \to \infty} P( X \geq s ) =0 ## and ## \lim_{s \to \infty} P( X \leq -s ) =0 ##:

Let ## (s_n)_{n=1}^\infty ## be a monotonically increasing sequence with ## \lim_{ n \to \infty } s_n = \infty ##. The sequences ## \{ X \geq s_n \}_{n=1}^\infty ## and ## \{ X \leq -s_n \}_{n=1}^\infty ## are decreasing sequences with zero intersection:

##\bigcap_{n=1}^{\infty}\left\{X \leq-s_n\right\} = \bigcap_{n=1}^{\infty}\left\{X \geq s_n\right\} = \emptyset ##,
hence from continuity of probability:

##
\begin{aligned}
&0=\mathbb{P}(\emptyset)=\mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\{X \geq s_n\right\}\right)=\lim _{n \rightarrow \infty} \mathbb{P}\left(X \geq s_n\right) \\
&0=\mathbb{P}(\emptyset)=\mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\{X \leq-s_n\right\}\right)=\lim _{n \rightarrow \infty} \mathbb{P}\left(X \leq-s_n\right)
\end{aligned}
##

Hence we'll deduce:

##
\lim _{s \rightarrow \infty} \mathbb{P}(|X| \geq s)=\lim _{s \rightarrow \infty}(\mathbb{P}(X \geq s)+\mathbb{P}(X \leq-s))=0
##
and we're finished.

Questions:
1. I understand that the proof above is according to Heine's definition of limit, but if so I don't understand why we took ## (s_n)_{n=1}^\infty ## to be a monotonically increasing sequence and not an arbitrary sequence? ( we'd also like to prove for sequences that do go to infinity but are not necessarily monotonically increasing ).
2. Why does the equation ## \mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\{X \geq s_n\right\}\right)=\lim _{n \rightarrow \infty} \mathbb{P}\left(X \geq s_n\right) ## hold? how did we go from the left side to the right side?

Thanks in advance for any help!

Questions:
1. I understand that the proof above is according to Heine's definition of limit, but if so I don't understand why we took ## (s_n)_{n=1}^\infty ## to be a monotonically increasing sequence and not an arbitrary sequence? ( we'd also like to prove for sequences that do go to infinity but are not necessarily monotonicallyincreasing ).

2. Why does the equation ## \mathbb{P}\left(\bigcap_{n=1}^{\infty}\left\{X \geq s_n\right\}\right)=\lim _{n \rightarrow \infty} \mathbb{P}\left(X \geq s_n\right) ## hold? how did we go from the left side to the right side?

What happens if $s_n$ is not necessarily strictly increasing? Since the minimum of an intersection is the maximum of the minima, we have $$\bigcap_{n=1}^N \{X \geq s_n\} = \{X \geq \max_{1 \leq n \leq N} s_n\}.$$ Define $$M_N = \max_{1 \leq n \leq N} s_n.$$ Then $M_n$ is an increasing sequence with $M_n \to \infty$. Then $$\begin{split} \mathbb{P}\left( \bigcap_{n=1}^\infty \{X > s_n\}\right) &= \lim_{N \to \infty} \mathbb{P}\left(\bigcap_{n=1}^N \{X \geq s_n\}\right) \\ &= \lim_{N \to \infty} \mathbb{P}(\{X \geq M_N\}).\end{split}$$

• CGandC
Thanks alot! everything's crystal clear now!

I also found an answer to my second question which stems from the law of continuity of probability which says the following ( in case anyone's interested ):
Let there be a monotonically decreasing sequence of events ## A_1 \supseteq A_2 \supseteq ... ## in probability space ## ( \Omega , \mathbb{P} ) ##. Then: ## \mathbb{P}\left( \bigcap_{n=1}^\infty A_n \right) = \lim_{n \to \infty } \mathbb{P}( A_n) ##