Suppose Y 1 , Y 2 , … is any sequence of iid real valued random variables with E ( Y 1 ) = ∞ . Show that, almost surely, lim sup n ( | Y n | / n ) = ∞ and lim sup n ( | Y 1 + . . . + Y n | / n ) = ∞.I have solved the first part by considering non-negative integer iid r.vs X n = floor ( | Y n | ) and using E ( X ) = ∑ 0 ∞ P ( X ≥ n ) then doing some clever tricks so I can apply the (2nd) Borel-Cantelli lemma, but I'm not really sure how I can use the same approach to solve the second part seeing as it is tempting to set S n = floor ( | Y 1 + . . . + Y n | ) but then the S i are not iid. I'm pretty sure its gonna be Borel-Cantelli again (since limsup) so I need to come up with the right events. Please can someone nudge me in the right direction.Hints only pleaseEDIT: Suppose lim sup n | a 1 + . . . + a n | / n &lt; ∞. Then set S n = ∑ 1 n a k | a n | n = | S n − S n − 1 | n ≤ | S n | n + | S n − 1 | n − 1 bounded

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Suppose       Y    1    ,      Y    2    ,  … is any sequence of iid real valued random variables with   E  (      Y    1    )  =  ∞ . Show that, almost surely,       lim sup    n    (      |        Y    n        |        /    n  )  =  ∞ and       lim sup    n    (      |        Y    1    +  .  .  .  +      Y    n        |        /    n  )  =  ∞.I have solved the first part by considering non-negative integer iid r.vs       X    n    =  floor  (      |        Y    n        |    ) and using   E  (  X  )  =      ∑    0    ∞    P  (  X  ≥  n  ) then doing some clever tricks so I can apply the (2nd) Borel-Cantelli lemma, but I&#039;m not really sure how I can use the same approach to solve the second part seeing as it is tempting to set       S    n    =  floor  (      |        Y    1    +  .  .  .  +      Y    n        |    ) but then the       S    i   are not iid. I&#039;m pretty sure its gonna be Borel-Cantelli again (since limsup) so I need to come up with the right events. Please can someone nudge me in the right direction.Hints only pleaseEDIT: Suppose       lim sup    n        |        a    1    +  .  .  .  +      a    n        |        /    n  &amp;lt;  ∞. Then set       S    n    =      ∑    1    n        a    k                      |                    a        n                    |              n    =                    |                    S        n            −              S                  n          −          1                            |              n      ≤                    |                    S        n                    |              n    +                    |                    S                  n          −          1                            |                    n      −      1      bounded

Hailee Henderson · Accepted Answer

Hint: If   (      a    n    ) is a sequence of real numbers with   lim  sup            |              a      1        +          a      2        +    ⋯    +          a      n              |            /    n  &amp;lt;  ∞ then   lim  sup                    |                    a        n                    |              n    &amp;lt;  ∞.

Suppose Y 1 </msub> , Y 2 </msub> , …

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