So my task is the following: Consider a sequence of random variables ( X n </ms

Alexander Ware

Alexander Ware

Answered question

2022-06-04

So my task is the following: Consider a sequence of random variables ( X n ) n N with X n a . s . X and X n B a.s. for B R . Show that X n L p X for all p N .

My ideas till now:
1. In a first step i tried to show that the expected value is bounded by both sides from zero: 0 E ( | X n X | p ) 0 Problem here is that i would need that | X n | B a.s.
2. My second attempt was to use the dominated convergence theorem. With Y n = B it follows that if Y = B that Y n a . s . Y and E ( | Y n Y | ). Now I also need that | X n | Y and then it follows that E ( | X n X | ) n 0.

Probably someone can give me a hint or another way of solving this.

Answer & Explanation

Kolten Bowen

Kolten Bowen

Beginner2022-06-05Added 3 answers

X n = n I ( 0 , 1 n ) , X = 0 , B = 0 , p = 1 on (0,1) with Lebesgue measure is a counter-example.
If | X n | B then E | X n X | p 0 for all p > 0 by DCT.

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