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Yanqi Qiu

Publications and source records attributed to Yanqi Qiu.

At least 37 records · Page 2Linked to original sources
arXiv2021

Boundedness of Gaussian random sums on trees

Let $\mathcal{T}$ be a rooted tree endowed with the natural partial order $\preceq$. Let $(Z(v))_{v\in \mathcal{T}}$ be a sequence of independent standard Gaussian random variables and let $α= (α_k)_{k=1}^\infty$ be a sequence of real numbers with $\sum_{k=1}^\infty α_k^2<\infty$. Set $α_0 =0$ and define a Gaussian process on $\mathcal{T}$ in the following way: \[ G(\mathcal{T}, α; v): = \sum_{u\preceq v} α_{|u|} Z(u), \quad v \in \mathcal{T}, \] where $|u|$ denotes the graph distance between the vertex $u$ and the root vertex. Under mild assumptions on $\mathcal{T}$, we obtain a necessary and sufficient condition for the almost sure boundedness of the above Gaussian process. Our condition is also necessary and sufficient for the almost sure uniform convergence of the Gaussian process $G(\mathcal{T}, α; v)$ along all rooted geodesic rays in $\mathcal{T}$.

math.PR↗
arXiv2021

The Patterson-Sullivan Interpolation of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces

The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $\mathbb{D}_d$ in $\mathbb{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and Theorem 4.11); optimal simultaneous uniform interpolation for weighted Bergman spaces (cf. Theorem 1.8, Proposition 1.9 and Theorem 4.13); strong simultaneous uniform interpolation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem 4.15); and establish the impossibility of the uniform simultaneous interpolation for the Bergman space $A^2(\mathbb{D}_d)$ on $\mathbb{D}_d$ (cf. Theorem 1.12 and Theorem 6.7).

math.CV↗
arXiv2021

Gaussian limit for determinantal point processes with $J$-Hermitian kernels

We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $\mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.

math.PR↗
arXiv2020

On a class of random sets of positive integers

In this note, we study a class of random subsets of positive integers induced by Bernoulli random variables. We obtain sufficient conditions such that the random set is almost surely lacunary, does not have bounded gaps and contains infinitely many arithmetic progressions, respectively.

math.PR↗
arXiv2020

The metric projections onto closed convex cones in a Hilbert space

We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr{H}$ generated by a sequence $\mathcal{V} = \{v_n\}_{n=0}^\infty$. The first main result of this paper provides a sufficient condition under which we can identify the closed convex cone generated by $\mathcal{V}$ with the following set: \[ \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in $\mathscr{H}$}\bigg\}. \] Then, by adapting classical results on general convex cones, we give a useful description of the metric projection of a vector onto $\mathcal{C}[[\mathcal{V}]]$. As applications, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with non-negative coefficients.

math.FA↗
arXiv2020

Rigidity of determinantal point processes on the unit disc with sub-Bergman kernels

We give natural constructions of number rigid determinantal point processes on the unit disc $\mathbb{D}$ with sub-Bergman kernels of the form \[ K_Λ(z, w) = \sum_{n\in Λ}(n+1) z^n \bar{w}^n, \quad z, w \in \mathbb{D}, \] with $Λ$ an infinite subset of the set of non-negative integers. Our constructions are given both in a deterministic method and a probabilisitc method. In the deterministic method, our proofs involve the classical Bloch functions.

math.PR↗
arXiv2020

Hyper-positive definite functions II: A complete study of branching-Toeplitz operators

We introduce and give a more or less complete study of a family of branching-Toeplitz operators on the Hilbert space $\ell^2(T_q)$ indexed by a rooted homogeneous tree $T_q$ of degree $q\ge 2$. The finite dimensional analogues of such operators form a very natural family of structured sparse matrices called branching-Toeplitz matrices and will also be investigated. The branching-Toeplitz operators/matrices in this paper should be viewed as natural generalizations of the standard Toeplitz operators/matrices. We will apply our results to construct a family of determinantal point processes on homogeneous trees which are branching-type strong stationary stochastic processes.

math.FA↗
arXiv2019

Patterson-Sullivan measures for point processes and the reconstruction of harmonic functions

The Patterson-Sullivan construction is proved almost surely to recover every harmonic function in a certain Banach space from its values on the zero set of a Gaussian analytic function on the disk. The argument relies on the slow growth of variance for linear statistics of the concerned point process. As a corollary of reconstruction result in general abstract setting, Patterson-Sullivan reconstruction of harmonic functions is obtained in real and complex hyperbolic spaces of arbitrary dimension.

math.PR↗
arXiv2019

Radial Toeplitz operators and invariant determinantal point processes on Cayley trees

We give a complete description of bounded radial Toeplitz operators on the Hilbert space associated with a Cayley tree. As an application, we give a complete classification of a rich family of determinantal point processes on Cayley trees whose correlation kernels and thus distributions are invariant under the action of the graph automorphisms of the Cayley trees.

math.FA↗
arXiv2019

Hyper-positive definite functions I: scalar case, branching-type stationary stochastic processes

We propose a definition of branching-type stationary stochastic processes on rooted trees and related definitions of hyper-positivity for functions on the unit circle and functions on the set of non-negative integers. We then obtain (1) a necessary and sufficient condition on a rooted tree for the existence of non-trivial branching-type stationary stochastic processes on it, (2) a complete criterion of the hyper-positive functions in the setting of rooted homogeneous trees in terms of a variant of the classical Herglotz-Bochner Theorem, (3) a prediction theory result for branching-type stationary stochastic processes. As an unexpected application, we obtain natural hypercontractive inequalities for Hankel operators with hyper-positive symbols.

math.PR↗
arXiv2019

Stationary determinantal processes: $ψ$-mixing property and $L^q$-dimensions

The results of this paper are 3-folded. Firstly, for any stationary determinantal process on the integer lattice, induced by strictly positive and strictly contractive involution kernel, we obtain the necessary and sufficient condition for the $ψ$-mixing property. Secondly, we obtain the existence of the $L^q$-dimensions of the stationary determinantal measure on symbolic space $\{0, 1\}^\mathbb{N}$ under appropriate conditions. Thirdly, the previous two results together imply the precise increasing rate of the longest common substring of a typical pair of points in $\{0, 1\}^\mathbb{N}$.

math.PR↗
arXiv2019

Spectral gap of sparse bistochastic matrices with exchangeable rows with application to shuffle-and-fold maps

We consider a random bistochastic matrix of size $n$ of the form $M Q$ where $M$ is a uniformly distributed permutation matrix and $Q$ is a given bistochastic matrix. Under mild sparsity and regularity assumptions on $Q$, we prove that the second largest eigenvalue of $MQ$ is essentially bounded by the normalized Hilbert-Schmidt norm of $Q$ when $n$ grows large. We apply this result to random walks on random regular digraphs and to shuffle-and-fold maps of the unit interval popularized in fluid mixing protocols.

math.DS↗
arXiv2018

Kernels of conditional determinantal measures and the proof of the Lyons-Peres Conjecture

The main result of this paper, Theorem 1.5, establishes a conjecture of Lyons and Peres: for a determinantal point process governed by a reproducing kernel, the system of kernels sampled at the particles of a random configuration is complete in the range of the kernel. A key step in the proof, Lemma 1.11, states that conditioning on the configuration in a subset preserves the determinantal property, and the main Lemma 1.12 is a new local property for kernels of conditional point processes. In Theorem 1.7 we prove the triviality of the tail sigma-algebra for determinantal point processes governed by self-adjoint kernels.

math.PR↗
arXiv2018

On Number Rigidity for Pfaffian Point Processes

Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of additive statistics in the spirit of Ghosh and Peres. Second, a sufficient condition for number rigidity of stationary Pfaffian processes, relying on the Kolmogorov criterion for interpolation of stationary processes and applicable, in particular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectral measure for additive statistics.

math.PR↗
arXiv2017

A rigidity property of superpositions involving determinantal processes

The main result of this paper states that if $(N, Π)$ is a pair of independent point processes on a common ground space with $N$ Poisson and $Π$ determinantal induced by a locally trace class (not necessarily self-adjoint) correlation kernel, then their independent superposition $N + Π$ determines uniquely the distributions of $N$ and $Π$.

math.PR↗