Semigroup inequalities, stochastic domination, Hardy's inequality, and strong ergodicity
For the classical L p -spaces of signed measures on N, we devise a framework in which bounds for a sub-Markovian semigroup of interest can be obtained, up to a constant factor, from bounds for another tractable semigroup that dominates stochastically the first one. The main tools are the Hardy inequality, the definition of related auxiliary L p spaces suited to take advantage of the domination, and the proof that the norms are equivalent to the classical ones if the reference measure is quasi-geometrically decreasing. We illustrate the results using birth-death and single-birth processes.
Functional central limit theorems for a large network in which customers join the shortest of several queues
Abstract.We consider N single server infinite buffer queues with service rate β. Customers arrive at rate Nα, choose L queues uniformly, and join the shortest. We study the processes  for large N, where RNt(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data RN0 satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the RN in equilibrium under the stability condition α<β, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits  lim N→∞ lim t→∞= lim t→∞ lim N→∞ by a compactness-uniqueness method. We deduce a posteriori the CLT for RN0 under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit.
Indeterminate symmetric moment problems
Let {n,(x)} be a sequence of orthogonal polynomials with respect to a real distribution &. According to a classical theorem of Reisz 191, (x,(x)] is complete in L: if and only if w is either the solution of a determined Hamburger moment problem or an extremal solution of an indeterminate moment problem. In the special case where the zeros of the orthogonal polynomials have a finite infimum a and the Hamburger moment problem is indeterminate, there is an extremal solution which is essentially the only solution whose spectrum is a subset of [a, co) [4]. This solution is, in a sense, the “best” solution from the viewpoint of orthogonal polynomials since its spectrum consists of the limit points of the zeros of the polynomials. Askey remarked to the author that in view of Riesz’ theorem, it is surprising that so little attention has been paid to the question of extremal solutions. (Moak [8] has recently studied the extremal solutions associated with the “q-Laguerre,” nee generalized Stieltjes-Wigert, polynomials.) Askey suggested it would be desirable to investigate the existence of “best” extremal solutions in the general case or, as a first step, even in the “symmetric case” (moments of odd order are 0). In this paper, we will consider this special case and the related systems of orthogonal polynomials. In particular, we will show that an indeterminate symmetric moment problem has precisely two “symmetric” extremal solutions. This result, which we found surprizing at first, appears after reflection quite natural (and perhaps obvious) since the two solutions are the symmetrizations of the “best” extremal solutions of the related moment problems on [0, co).