Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non overlapping hyperspheres whose centers are the closest to $p$ under the Euclidean metric (although I’m eventually interested in exploring other metrics as … Read more

regularity of zero point

We consider 1-d process X X(t)=bt+Jt+Mt where b is constant, M is a continuous martingale process with M(0)=0, and J is a symmestric α-stable process with its Levy symbol η(u)=−|u|α for some constant α∈[1,2). Its Levy measure is ν(y)=1|y|1+α. Consider τ=inf I want to know if the following claim is true. [Claim] \tau = 0 … Read more

What is the influence of unreliable comparisons on the results of sorting

Considering sorting algorithms based solely on binary comparisons of the elements to be sorted(algorithms such as insertion sort, selection sort, quicksort, and so on), what problems do we face when those comparisons are unreliable? We assume the elements of the sequence x=(x1,x2,…,xn) to be sorted are distinct. We assume further that the only comparisons subject … Read more

Interplay between CLT and convergence in Total Variation

Given a random variable $X$ with bounded moments such that $E[X] = 0, E[X^2] = 1$, let $F_n$ denote the distribution $\sum\limits_{i=1}^d\frac{X_i}{\sqrt{n}}$ where each $X_i$ is an independent copy of $X$. It is known that as $n \to \infty$, $F_n$ converges in distribution to a standard Gaussian. The Berry-Esseen inequality then gives us a quantitative … Read more

An inequality involving conditional variance and its connection to information theory

Given absolutely continuous random variables (X,Y) with joint distribution PXY, we construct Z:=√γY+NG where NG∼N(0,1) and is independent of (X,Y) and γ≥0. The conditional variance of Y given Z is defined as var(Y|Z)=E[(Y−E[Y|Z])2|Z], and hence E[var(Y|Z)]=E[(Y−E[Y|Z])2]. What are known about conditional variance: Independence: If Y is independent of Z (i.e., γ=0), then E[var(Y|Z)]=var(Y) and if … Read more

Self-adjusting random walk

Let Xt be a random process such that X1=0Xt=Xt−1+{At,Xt−1≥0Bt,Xt−1<0 where At and Bt are i.i.d. random variables with the properties E[At]=−α,Var[At]=σ2E[Bt]=α,Var[Bt]=σ2. Question: I am looking for techniques to establish an upper bound on E[|Xt|] or E[X2t] which holds for all t∈N in terms of α and σ. I can write out E[X2t] as follows E[X2t]=E[(Xt−1+1Xt−1≥0At+1Xt−1<0Bt)2]=E[X2t−1]+E[(1Xt−1≥0At+1Xt−1<0Bt)2]+2E[Xt−1(1Xt−1≥0At+1Xt−1<0Bt)]=E[X2t−1]+α2+σ2−2E[|Xt−1|]α … Read more

concentration of functions of Gaussian processes

Let C∈Rn be a subset of the unit ball. Also let a1,a2,…,am∈Rn be i.i.d. random Gaussian vectors N(0,I) then by Gordon’s lemma we know supy∈C|1mm∑k=1(aTky)2−‖ holds with high probability as long as \begin{align*} m\ge c\frac{\omega^2(\mathcal{C})}{\delta^2}, \end{align*} for a fixed numerical constant c (in fact I think c is at most 9). Here \omega(\mathcal{C}) is the … Read more

Short time asymptotics for Brownian motion on a compact manifold

Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in this context, quasi-isometric means that $c_1 g_{\mathbb{R}^n} \leq g \leq c_2 g_{\mathbb{R}^n}$ on $B(p, r)$ and $B(q, s)$. Now, consider a compact … Read more

Concentration of infinite-dimensional Gaussian measure

I have the question about finding the subspace of concentration of a Gaussian Measure. More precisely: $\textbf{Question:}$ Assume we have a separable Hilbert space $\ell_2$ with Borel $\sigma$-algebra $B(\ell_2)$ and a centered Gaussian measure $\mu$, i.e. we consider the probability space $(\ell_2, B(\ell_2), \mu)$. There is also a linear continuous operator $Q:\ell_2 \rightarrow (\mathbb{R}^{\infty},\rho)$, where … Read more

Random polyominoes containing 2×22\times2 squares

The construction quoted in the question “How to sample a uniform random polyomino?” claims to produce a “uniform random polyomino”. But apart from the mentioned possibility of getting stuck, it also seems to me that the polyominoes obtained by this construction tend to contain less 2×2 squares than average. This is because if three cells … Read more