Example where X is not in the Lebesgue linear space.
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...
I'm trying to find a counterexample where \lim_{n \to +\infty} P(|X|>n) = 0 but X \notin L where L is the lebesgue linear space.
∫|X|I(|X|>n)dp + ∫|X|I(|X|≤n)dp = ∫|X|dp therefore
∫nI(|X|>n)dp + ∫|X|I(|X|)dp ≤ ∫|X|dp
Suppose ∫I(|X|>n)dp = 1/(n ln n)
Clearly the hypothesis is satisfied...
That's exactly right. Thank you for pointing that out. I corrected the question.
The reason for the repeats is the following:
I'm preforming a symmetrization on the set K and the algorithm is such that it produces the above sequence.
I got confused myself because once I got the...
Conjecture Regarding Rotation of a Set by a Sequence of Angles.
Consider the following sequence, where the elements are rational numbers mulriplied by \pi:
(\alpha_{i}) = \hspace{2 mm}\pi/4,\hspace{2 mm} 3\pi/8,\hspace{2 mm} \pi/4,\hspace{2 mm} 3\pi/16,\hspace{2 mm} \pi/4,\hspace{2 mm}...
Yes for sure!
I actually solved this problem a while ago! It turns out the conjecture doesn't hold and the counterexample is the van der corput sequence.
Def: A low discrepancy sequence is a uniformly distributed sequence with minimal discrepancy, O(logN/N).
Question: Let <x> denote the fractal part of an irrational number x. Let (<x_n>) be an arbitrary low discrepancy sequence. Is it always true that :
\lim_{n \to +\infty}|<x_n - x_{n-1}> -...