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\(\sup_n X_n\) is not a number but a function from the sample space to reals (plus infinity)...Of course, setting $\displaystyle \text{sup}_{n} X_{n}=B$, for A>B the 'infinite sum' vanishes.
I'm afraid that the question has been wrongly expressed and in particular there is confusion between the random variables $X_{n}$ and the probabilities $P_{n}= P\{X_{n}>A\}$...\(\sup_n X_n\) is not a number but a function from the sample space to reals (plus infinity)...
Thank you for everybody who replied to my post, special thanks to girdav for the hint and to Evgeny.Makarov for 'protecting' my question.Hint: use Borel-Cantelli lemma.
Thank you so much for your quick and clear answer. I really appreciate it!I meant this result:.
If we assume that for some $A>0$ the series $\sum_{n\geq 0}P(X_n\geq A)$ is convergent then by Borel-Cantelli lemma $P(\limsup_n X_n\geq A)=0$ . Now assume that for all $A$ we have that $\sum_{n\geq 0}P(X_n\geq A)$ is divergent, and apply a converse of Borel-Cantelli lemma, which works for independent random variables.
I'm afraid your wording is as wrong as ever. You're always putting some weird notations and it's quite obvious that you're not used to using common things in probability. Hence the probability that you understood wrongly is superior to the probability of the question being wrongly worded.I'm afraid that the question has been wrongly expressed and in particular there is confusion between the random variables $X_{n}$ and the probabilities $P_{n}= P\{X_{n}>A\}$...
Kind regards
$\chi$ $\sigma$
I'm afraid... following Dante Alighieri's sentence reported in the signature... that I'm no time to waste with monkeys ...I'm afraid your wording is as wrong as ever...
A monkey that can probably speak a better English than yours, but that wouldn't bother looking for the translation of an Italian sentence no one cares about. And I did mean wording, not working.I'm afraid... following Dante Alighieri's sentence reported in the signature... that I'm no time to waste with monkeys ...
Kind regards
$\chi$ $\sigma$