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Feb
1
accepted Let $x_{1},…,x_{n}\in\mathbb{Z}_{\leq1}$ with sum $1$. There is a circulant permutation s.t $\sum_{i=1}^{k}x_{\pi(i)}\leq0 \forall k\leq n-1$.
Feb
1
comment Let $x_{1},…,x_{n}\in\mathbb{Z}_{\leq1}$ with sum $1$. There is a circulant permutation s.t $\sum_{i=1}^{k}x_{\pi(i)}\leq0 \forall k\leq n-1$.
@ByronSchmuland I'm not familiar with that result I'm afraid. Could you provide a reference please? Preferably one with a proof of the lemma. Thanks!
Feb
1
revised Let $x_{1},…,x_{n}\in\mathbb{Z}_{\leq1}$ with sum $1$. There is a circulant permutation s.t $\sum_{i=1}^{k}x_{\pi(i)}\leq0 \forall k\leq n-1$.
added 14 characters in body; edited title
Feb
1
asked Let $x_{1},…,x_{n}\in\mathbb{Z}_{\leq1}$ with sum $1$. There is a circulant permutation s.t $\sum_{i=1}^{k}x_{\pi(i)}\leq0 \forall k\leq n-1$.
Jan
31
comment Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.
$S_{n}^{'}$ is just a tail of $S_{n}$ that starts after the first hit so if $S_{n}^{'}$ hits the interval again so does the original $S_{n}$, this is why it seems logical to me to just restart and reapply the same argument after each hit. Where is the flaw in the logic?
Jan
31
comment $X_{n}$ independent and there is $a_{n}\to0$ s.t $\lim\limits _{m\to\infty}a_{m}\sum_{n=1}^{m}X_{n}$ is finite w.p 1. Then the limit is constant.
@SangchulLee The assumption tells me that the limit exists and is finite a.s but not that it is always the same (i.e) constant which is what I'm trying to show. In order to use the argument you suggest it seems to me I need to know that the value of the limit is also bounded in order to make sure that I can start with an interval that contains the limit a.s. Why is this true?
Jan
31
comment Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.
@Dominik If I assume by contradiction that $S_{n}$ doesn't change sign and it's not fixed at $0$ then it's either always positive or always negative, I just chose one of those cases, the other case would be proven in the same way. You are right though about the fact that a sequence of strictly positive numbers can have limit equaling zero. If I showed that $S_{n}\geq c>0$ instead and that would have sorted it out. What about the logic that followed later, if I do show that $S_{n}$ changes sign is the rest correct?
Jan
31
comment $X_{n}$ independent and there is $a_{n}\to0$ s.t $\lim\limits _{m\to\infty}a_{m}\sum_{n=1}^{m}X_{n}$ is finite w.p 1. Then the limit is constant.
@SangchulLee What sort of bisection will work though? I need a bisection that will gradually be reduced to a singleton.
Jan
31
revised Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.
Added proof attempt
Jan
31
accepted Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?
Jan
31
asked Let $\{ X_{n}\} _{n\geq1}$ be IID s.t $\mathbb{E}[X_{i}]=0$ and $|X_{i}|\leq K$. Show $S_{n}$ visits $[-K,K]$ infinitely often.
Jan
30
comment Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?
Just as you were responding I thought about orthonogonal matrices.. How silly of me...
Jan
30
asked Suppose $\Vert Ax\Vert _{2}=\Vert Bx\Vert _{2}$ for all $x\in\mathbb{R}^{n}$ , does that imply $A=B$ or $A=-B$?
Jan
17
accepted Proof of Hoeffding's Covariance Identity
Jan
17
asked Proof of Hoeffding's Covariance Identity
Jan
16
comment Clarification of notation in multivariate taylor expansion
$\nabla^{2}f\left(x+tp\right)p$ as a function of $t$ is a function from $\mathbb{R}$ to $\mathbb{R}^{n}$ . What exactly does it mean to integrate such a function. Does it mean to integrate each coordinate of the function and return the vector of integrated results?
Jan
16
asked Clarification of notation in multivariate taylor expansion
Jan
16
accepted Dense subsets of the metric space of measurable sets with metric $d\left(A,B\right)=\mu\left(A\triangle B\right) $
Jan
10
asked Property of atomless probability measures
Jan
10
asked Dense subsets of the metric space of measurable sets with metric $d\left(A,B\right)=\mu\left(A\triangle B\right) $