Recent content by logarithmic

Everything here is in a Hilbert space. If x_n\to x and y_n\to y in norm, then under what conditions does <x_n,y_n>\to <x,y>? Is this always true, and why? Does anyone have a source?

Let C_b^\infty(\mathbb{R}^n) be the space of infinitely differentiable functions f, such that f and all its partial derivatives are bounded. Is C_b^\infty(\mathbb{R}^n) dense in L^2(\mathbb{R}^n)? I think the answer is yes, because C_b^\infty(\mathbb{R}^n) contains C_0^\infty(\mathbb{R}^n), the...

Suppose that x\in H, where H is a Hilbert space. Then x has an orthogonal decomposition x = \sum_{i=0}^\infty x_i. I have a linear operator P (more specifically a projection operator), and I want to write: P(x) = \sum_{i=0}^\infty P(x_i). How can I justify taking the operator inside the...

Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism. If X is a dense set in H, then is A(X) a dense set in K? Any references to texts would also be helpful.

The answer seems to obviously be yes. But it's not so obvious to show it. I'm working with random variables. So the L^2 norm of X is E(X^2)^{1/2}, where E is the expected value. Thus, we want to show: if E((X_n-X)^2)\to0, then E(X_n^2)\to E(X^2). From E((X_n^2-X)^2)\to0, we get...

https://en.wikipedia.org/wiki/Convergence_of_random_variables#Definition_3

If the PDF converges, that's convergence in distribution, which is weaker than almost sure convergence.

Yes, X is a random variable, not a number. I'm not sure if looking at the MGF will help though, as the only results I know about MGF and convergence (e.g. Levy's continuity theorem, and the theorem that if E(X_n^p)\to E(X^p) for all p, then we have convergence in distribution) are about...

I'm trying to prove that if \{X_n\} is independent and E(X_n)=0 for all n, and \sum_{n}E(X_n^2) <\infty, then \sum_{n}X_n converges almost surely. What I've got so far is the following: Denote the partial sums by \{S_n\}, then proving almost sure convergence is equivalent to showing that...

e.v. means eventually. For sets S_n, \{S_n e.v.\}:= \cup_{n\in \mathbb{N}}\cap_{m\geq n}S_n.

Thanks. How would we show that P(\liminf \bar{X}_n < c) = 1. I think this is P(\bar{X}_n < c \text{ e.v.}) = 1 - P(\bar{X}_n > c \text{ i.o.}) = 1 - 1 = 0, not 1.

I don't think there is a paradox, as the argument using ">" is valid. It shows that in the link, and the conclusion that the limsup is infinite is definitely correct. Something is wrong with it when I change to "<". Your argument doesn't make sense. Why can't we have a probability that contains...

Borel-Cantelli doesn't require independence. https://en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma The line above "Example". And I don't think what you're saying is true, as it would seem to imply that Borel-Cantelli can't be used on sample means, when it's used to prove the Strong Law of...

What about the following argument. Let X_i be standard Cauchy. It's a well known fact about the Cauchy distribution that \bar{X}_n has the same distribution as X_1, i.e. the sample mean of standard Cauchy is standard Cauchy, for any n. Let c \geq 0 be arbitrary. Now P(\bar{X}_n > c) = P(X_1 >...

On example 2 on page 2 of this link (http://math.arizona.edu/~jgemmer/bishopprobability3.pdf) it uses Borel-Canetelli for random variables in the same way I did, even though they've defined io for sets only.