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mehr1methanol
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Example where X is not in the Lebesgue linear space.
I'm trying to find an example where \lim_{n \to +\infty} P(|X|>n) = 0 but X \notin L where L is the Lebesgue linear space.
Relevant equations:
X is a random variabel, P is probability. I is indicator function.
The attempt at a solution
∫|X|I(|X|>n)dp + ∫|X|I(|X|≤n)dp = ∫|X|dp
∫nI(|X|>n)dp + ∫|X|I(|X|)dp ≤ ∫|X|dp
Suppose ∫I(|X|>n)dp = 1/(n ln n)
Clearly the hypothesis is satisfied because \lim_{n \to +\infty} P(|X|>n) = \lim_{n \to +\infty} ∫I(|X|>n)dp = \lim_{n \to +\infty} 1/( ln n) = 0
But I'm not sure how to conclude ∫|X|dp = ∞
Homework Statement
I'm trying to find an example where \lim_{n \to +\infty} P(|X|>n) = 0 but X \notin L where L is the Lebesgue linear space.
Relevant equations:
X is a random variabel, P is probability. I is indicator function.
The attempt at a solution
∫|X|I(|X|>n)dp + ∫|X|I(|X|≤n)dp = ∫|X|dp
∫nI(|X|>n)dp + ∫|X|I(|X|)dp ≤ ∫|X|dp
Suppose ∫I(|X|>n)dp = 1/(n ln n)
Clearly the hypothesis is satisfied because \lim_{n \to +\infty} P(|X|>n) = \lim_{n \to +\infty} ∫I(|X|>n)dp = \lim_{n \to +\infty} 1/( ln n) = 0
But I'm not sure how to conclude ∫|X|dp = ∞
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