Ergodic degrees for continuous-time Markov chains

This paper studies the existence of the higher orders deviation matrices for continuous time Markov chains by the moments for the hitting times. An estimate of the polynomial convergence rates for the transition matrix to the stationary measure is obtained. Finally, the explicit formulas for birth-death processes are presented.     

The Perron–Frobenius theorem – a proof with the use of Markov chains

The Perron–Frobenius theorem for an irreducible nonnegative matrix is proved using the matrix graph and the ergodic theorem of the theory of Markov chains. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 5–16.     

The Hyers theorem via the Markov–Kakutani fixed point theorem

We present an application of the Markov–Kakutani common fixed point theorem to the theory of stability of functional equation by proving some version of the Hyers theorem concerning approximate homomorphisms.     

The Carathéodory-Rashevsky-Chow theorem for the nonholonomic Lipschitz distributions

It is proved that if a k-dimensional Lipschitz distribution H in ? ^k+1 is nonholonomic in a connected domain, then every pair of points can be joined by an H-polygonal path.     

Idempotent ultrafilters,multipleweak mixing and Szemerédi’s theorem for generalized polynomials

It is possible to formulate the polynomial Szemerédi theorem as follows: Let q _i (x) ∈ Q[x] with q _i (Z) ⊂ Z, 1 ≤ i ≤ k. If E ⊂ N has positive upper density, then there are a, n ∈ N such that

The central limit theorem for Chébli-Trimèche hypergroups

Let (S _nn>-1) be a random walk on a hypergroup ( ₊ , *), i.e., a Markov chain with transition kernelN(x, A) = _x * (A), where is a fixed probability measure on ₊ such that the second moment exists. Then depending on the growth of the hypergroup two situations can occur: when ( ₊ , *) is of exponential growth then it is shown thatS _n is asymptotically normal. In the case of polynomial growth {more precisely, if the densityA of the Haar measure of ( ₊ , *) satisfies lim[A()/A()]=}, the normalized variablesS _n/[n Var()/(+1)]^1/2 converge to a Rayleigh distribution with parameter .     

The general dévissage theorem for Witt groups of schemes

Let be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex then there exists a dualizing complex of Z such that there is an isomorphism of coherent Witt groups for all . Received: 3 March 2006     

The Julia-Wolff-Carathéodory theorem in polydisks

The classical Julia-Wolff-Carathéodory theorem gives a condition ensuring the existence of the non-tangential limit of both a bounded holomorphic function and its derivative at a given boundary point of the unit disk in the complex plane. This theorem has been generalized by Rudin to holomorphic maps between unit balls inC ⁿ and by the author to holomorphic maps between strongly (pseudo)convex domains. Here we describe Julia-Wolff-Carathéodory theorems for holomorphic maps defined in a polydisk and with image either in the unit disk, or in another polydisk, or in a strongly convex domain. One of the main tools for the proof is a general version of the Lindelöf principle valid for not necessarily bounded holomorphic functions.     

Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.     

Strong stationary duality for M?bius monotone Markov chains

For Markov chains with a finite, partially ordered state space, we show strong stationary duality under the condition of M?bius monotonicity of the chain. We give examples of dual chains in this context which have no downwards transitions. We illustrate general theory by an analysis of nonsymmetric random walks on the cube with an interpretation for unreliable networks of queues.     

On Cramér's theorem for capacities

In this Note, our aim is to obtain Cramér's upper bound for capacities induced by sublinear expectations.     

The Gauss–Wilson theorem for quarter-intervals

We define a Gauss factorial N _n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!$ for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of  $\frac{p-1}{4}!$ modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}$ . We also report on computations that were required in the process.     

State-dependent Foster–Lyapunov criteria for subgeometric convergence of Markov chains

We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster–Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in Connor and Kendall (2007) [2] concerning the existence of so-called ‘tame’ Markov chains. Furthermore, we show that our ‘subsampled drift condition’ implies the existence of finite moments for the return time to a small set.     

A generalized min–max theorem for functionals of strongly indefinite sign

We consider non-linear Schrödinger equations of the following type: $$\begin{aligned} \left\{ \begin{array}{l} -\Delta u(x) + V(x)u(x)-q(x)|u(x)|^\sigma u(x) = \lambda u(x), \quad x\in \mathbb{R }^N \\ u\in H^1(\mathbb{R }^N)\setminus \{0\}, \end{array} \right. \end{aligned}$$ where $N\ge 1$ and $\sigma >0$ . We will concentrate on the case where both $V$ and $q$ are periodic, and we will analyse what happens for different values of $\lambda $ inside a spectral gap $]\lambda ^-,\lambda ^+[$ . We derive both the existence of multiple orbits of solutions and the bifurcation of solutions when $\lambda \nearrow \lambda ^+$ . Thereby we use the corresponding energy function ${I_\lambda }$ and we derive a new variational characterization of multiple critical levels for such functionals: in this way we get multiple orbits of solutions. One main advantage of our new view on some specific critical values $c_0(\lambda )\le c_1(\lambda )\le \cdots \le c_n(\lambda )\le \cdots $ is a multiplicity result telling us something about the number of critical points with energies below $c_n(\lambda )$ , even if for example two of these values $c_i(\lambda )$ and $c_j(\lambda )$ ( $0\le i<j\le n$ ) coincide. Let us close this summary by mentioning another main advantage of our variational characterization of critical levels: we present our result in an abstract setting that is suitable for other problems and we give some hints about such problems (like the case corresponding to a Coulomb potential $V$ ) at the end of the present paper.     

The central limit theorem for stationary Markov processes with normal generator—with applications to hypergroups

We extend the central limit theorem for additive functionals of a stationary, ergodic Markov chain with normal transition operator due to Gordin and Lif?ic, 1981 [A remark about a Markov process with normal transition operator, In: Third Vilnius Conference on Probability and Statistics 1, pp. 147–48] to continuous-time Markov processes with normal generators. As examples, we discuss random walks on compact commutative hypergroups as well as certain random walks on non-commutative, compact groups.     

Fejér's theorem for the classesV p

La successione di elementi di una serie di Fourier derivata di una funzione appartenente alla classe di WienerV _p,p>1 (rispettivamente alla classe di Waterman {n ^β}B V o à quella di ChanturiyaV [n ^β] per qualsiasi 0<β<1) è sommabile verso (f(x+0)?f(x?0))/π mediante i metodi di Cèsaro di ordine α>1?1/p (α>β).     

Asymptotic normality of the Parzen–Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt, Proc Natl Acad Sci USA 42:43–47, 1956 and Parzen, Ann Math Stat 33:1965–1976, 1962) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg’s method rather than on Bernstein’s small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the strong mixing coefficients which do not depend on the bandwidth.