I Prove that the tail of this distribution goes to zero

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The discussion centers on proving that the tail of a distribution approaches zero, specifically that the limit of the probability of a random variable exceeding a threshold goes to zero as the threshold approaches infinity. The proof utilizes a monotonically increasing sequence to demonstrate that both the upper and lower tails of the distribution converge to zero, leveraging the continuity of probability. Questions arise regarding the necessity of using a monotonically increasing sequence and the transition from the intersection of events to the limit of their probabilities. Clarifications indicate that even non-monotonically increasing sequences can be handled by considering the maximum of the sequence, which still leads to the desired conclusion. The law of continuity of probability is highlighted as a key principle supporting the proof.
CGandC
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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!
 
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CGandC said:
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 <br /> \bigcap_{n=1}^N \{X \geq s_n\} = \{X \geq \max_{1 \leq n \leq N} s_n\}. Define <br /> M_N = \max_{1 \leq n \leq N} s_n. Then M_n is an increasing sequence with M_n \to \infty. Then <br /> \begin{split}<br /> \mathbb{P}\left( \bigcap_{n=1}^\infty \{X &gt; s_n\}\right) &amp;= \lim_{N \to \infty} \mathbb{P}\left(\bigcap_{n=1}^N \{X \geq s_n\}\right) \\<br /> &amp;= \lim_{N \to \infty} \mathbb{P}(\{X \geq M_N\}).\end{split}
 
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) ##
 
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