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For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$: Question1:Is there always a nonsingular matrix $P$ over the same field $F$ whic …

The definition/notion of independence is always a bit odd in measure theoretic probability theory. Definition Given a probability space $(\Omega,\mathcal{F},P)$, two sets $A,B\in\mathcal{F}$ are d …

This post is partly inspired by Fourier Coefficients and Hölder Continuity. Typical proofs of the nowhere differentiability of Brownian paths is by contradiction using binary expansion from real anal …

It is known(Enumeration of graphs with a given and bounded degree sequence) that there is no a closed form formula for number of (labeled) graphs with bound on degree of vertices. Thus what I want to …

This is a graduate-level measure theory problem. I have thought throught it and asked on math.SE but received no satisfying answer. On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the …

An ensemble corresponding to a probability distribution usually has finite free energy so it is not a great loss of generality to assume that the ensemble also has finite energy in following discussio …

Given a clique complex $K$ constructed from a discrete set of vertices (i.e. its faces are isomorphic to the set of cliques in the 1-skeleton of $K$.), it seems that the Betti numbers $\beta_k$ define …

The Berry-Essen bound stated that $$\sup _{{x\in {\mathbb R}}}\left|\widehat{F_{n}(x)}-\Phi (x)\right|\leq C_{0}\cdot \psi _{0}$$ where $\psi _{0}(n)={\Big (}{\textstyle \sum \limits _{{i=1}}^{n}\si …

This post is partly inspired by this post. Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix While it is well-known that two basic cat …

Given a convex cone $P\subset X$ where $X$ is a $K$-vector space, $K=\mathbb{R}\text{ or }\mathbb{C}$ is a field. Suppose that $P$ satisfies positive element stipulations. (1) $X=P-P$. (2) $P\cap-P …

This question is partially inspired by the following MO post: What are some of the surprising results of finite sample statistical estimation? and current heated research front of high dimensional sta …

Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the solutio …

Due to the supporting hyperplane theorem, a convex set $C$ in a separable topological space has supporting hyperplance at each of its boundary points. The theorem only guarantees its existence, now I …

Given a probability model $\mathcal{P}=\{P_{\theta},\theta \in \Theta \}$ dominated by a $\sigma$-finite measure $\lambda$ (e.g. Lebesgue measure) on a locally compact space $\cal{X}$ along with $\sig …

I am now doing an article survey on the field of information geometry started by S.Amari and Barndorff-Nielson. I want to know some research situation in this field. I have read (4) and parts of (3). …