This paper studies beta ensembles on the real line in a high temperature regime, that is, the regime where βN→ const∈ (0 , ∞) , with N the system size and β the inverse temperature. For the global behavior, the convergence to the equilibrium measure is a consequence of a recent result on large deviation principle. This paper focuses on the local behavior and shows that the local statistics around any fixed reference energy converges weakly to a homogeneous Poisson point process.

We establish central limit theorems for general functionals on binomial point processes and their Poissonized version, which extends the results of Penrose–Yukich (Ann. Appl. Probab. 11(4), 1005–1041 (2001)) to the inhomogeneous case. Here functionals are required to be strongly stabilizing for add-one cost on homogeneous Poisson point processes and to satisfy some moments conditions. As an application, a central limit theorem for Betti numbers of random geometric complexes in the subcritical regime is derived.

The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter β is allowed to vary with the matrix size n. In particular, we show that as n→ ∞ with nβ→ ∞, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.

We establish the strong law of large numbers for Betti numbers of random ech complexes built on RN-valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on RN and the case where it is supported on a C1 compact manifold of dimension strictly less than N. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to Lp spaces, for all 1p<.

We study the limiting behavior of Gaussian beta ensembles in the regime where βn= const as n→ ∞. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by Benaych-Georges and Péché (J Stat Phys 161(3):633–656, 2015) with the explicit form of the intensity measure.

The paper studies the limiting behaviour of spectral measures of random Jacobi matrices of Gaussian, Wishart and MANOVA beta ensembles. We show that the spectral measures converge weakly to a limit distribution which is the semicircle distribution, Marchenko-Pastur distributions or Kesten-McKay distributions, respectively. The Gaussian fluctuation around the limit is then investigated.

The persistent homology of a stationary point process on RN is studied in this paper. As a generalization of continuum percolation theory, we study higher dimensional topological features of the point process such as loops, cavities, etc. in a multiscale way. The key ingredient is the persistence diagram, which is an expression of the persistent homology. We prove the strong law of large numbers for persistence diagrams as the window size tends to infinity and give a sufficient condition for the support of the limiting persistence diagram to coincide with the geometrically realizable region. We also discuss a central limit theorem for persistent Betti numbers.

Spectral measures of Wigner matrices are investigated. The Wigner semicircle law for spectral measures is proved. Regard this as the law of large number, the central limit theorem moments of spectral measures is also derived. The proof is based on moment method and combinatorial method.

An explicit formula for the mean spectral measure of a random Jacobi matrix is derived. The matrix can be regarded as the limit of Gaussian beta ensemble (G beta E) matrices as the matrix size N tends to infinity with the constraint that N beta is a constant.

The indicator function of the set of k-th power free integers is naturally extended to a random variable X(k)({dot operator}) on (ℤ○,λ), where ℤ○ is the ring of finite integral adeles and λ is the Haar probability measure. In the previous paper, the first author noted the strong law of large numbers for {X(k)({dot operator}+n)}∞n=1, and showed the asymptotics: Eλ[(Y(k)N)2]{equivalent to}1 as N→∞, where Y(k)N(x):=N-1/2k∑Nn=1(X(k)(x+n)-1/ζ(k)). In the present paper, we prove the convergence of Eλ[(Y(k)N)2]. For this, we present a general proposition of analytic number theory, and give a proof to this.

The paper deals with convergence of the Fourier series of q-Besicovitch almost periodic functions of the form f(t) similar to Sigma(infinity)(m=1)a(m)e(-i lambda mt)
where {lambda m} is a Dirichlet sequence, that is, a strictly increasing sequence of nonnegative numbers tending to infinity. In particular, we show that, for 1 &lt; q &lt; a, the Fourier series of f(t) converges in norm to the function f(t) itself with usual order, which is analogous to the convergence in norm of the Fourier series of a function on [0, 2 pi]. A version of the Carleson-Hunt theorem is also investigated.

This paper deals with a general Dirichlet series of the form
Sigma(infinity)(m=1) a(m)e(-lambda ms), s = sigma + it is an element of C,
where a(m) is an element of C, and {lambda(m)} is a strictly increasing sequence of nonnegative numbers tending to infinity. Let A be a subgroup of Rd, the real line with discrete topology, generated by (lambda(m)). The dual group of A is a compact group (Lambda) over cap with the normalized Haar measure v. Let x(lambda) be a character on (Lambda) over cap defined by x(lambda)(x) = x(lambda)(lambda is an element of A, x is an element of (Lambda) over cap). Then {x(lambda m)} is an orthonormal system in L-2((Lambda) over cap, nu). Thus, for any square summable sequence (a(m)), that is,
Sigma(infinity)(m=1) vertical bar a(m)vertical bar(2) &lt; infinity,
the series
Sigma(infinity)(m=1) a(m) chi lambda(m)
converges in L-2((Lambda) over cap, nu). Our main result claims that this series actually converges almost everywhere (with respect to the Haar measure nu). This result is analogous to Carleson's theorem for Fourier series and has some interesting consequences.

Let X (k)(n) be the indicator function of the set of k-th power free integers. In this paper, we study refinements of the density theorem, ζ being the Riemann zeta function. The method we take here is a compactification of ℤ; we extend S (k)N to a random variable on a probability space (ℤ̂, λ) in a natural way, where Ẑ is the ring of finite integral adeles and λ is the shift invariant normalized Haar measure. Then we investigate the rate of L 2-convergence of S (k)N, from which the above asymptotic result is derived.

In this paper, we show that the ring of finite integral adeles, together with its Borel field and its normalized Haar measure, is an appropriate probability space where limit-periodic arithmetical functions can be extended to random variables. The natural extensions of additive and multiplicative functions are studied. Besides, the convergence of Fourier expansions of limit-periodic functions is proved.

This paper deals with an index-2 notion for linear implicit difference equations (LIDEs) and with the solvability of initial value problems (IVPs) for index-2 LIDEs. Besides, the cocycle property as well as the multiplicative ergodic theorem of Oseledets type are also proved. (C) 2010 Elsevier Inc. All rights reserved.

This paper deals with the solvability of initial-value problem and with Lyapunov exponents for linear implicit random difference equations, i.e. the difference equations where the leading term cannot be solved. An index-1 concept for linear implicit random difference equations is introduced and a formula of solutions is given. Paper is also concerned with a version of the multiplicative theorem of Oseledets type.