Consider a sequence of identically distributed random variables ( X n </msub>

Emanuel Keith

Emanuel Keith

Answered question

2022-06-14

Consider a sequence of identically distributed random variables ( X n ), with E | X n | finite. Define Y n = sup ( | X 1 | , , | X n | ) / n. I must prove that Y n converges to 0 in L 1 .
If Ω is the domain of the random variables, then n Y n ( ω ) x if, and only if, | X i | ( ω ) x / n for i = 1 , , n. Thus, Y n 1 ( ( , x ] ) = i = 1 n X i 1 ( ( , x / n ] ). Then P ( Y n x ) = P ( i = 1 n X i 1 ( ( , x / n ] ) ). Does it follow from this that Y n is identically distributed to some of the | X i | ? Are there any hypothesis that may be missing?
Edit: there was a typo in the original question

Answer & Explanation

lisicw2

lisicw2

Beginner2022-06-15Added 11 answers

Let ε > 0. From the inequality
sup { | X 1 | , , | X n | } 1 { 1 i n : | X i | > n ε } i = 1 n | X i | 1 { | X i | > n ε }
and the fact that the X i 's are identically distributed, we have
E [ Y n ] = E [ Y n 1 { 1 i n : | X i | n ε } ] + E [ Y n 1 { 1 i n : | X i | > n ε } ] ε + E [ | X 1 | 1 { | X 1 | > n ε } ] .
Thus, because E [ | X 1 | ] < ,
lim sup n E [ Y n ] ε .
This is true for all ε > 0, so we can conclude that Y n 0 in L 1 .

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