Drift rate control of a Brownian processing system

A system manager dynamically controls a diffusion process Z that lives in a finite interval [0,b]. Control takes the form of a negative drift rate \theta that is chosen from a fixed set A of available values. The controlled process evolves according to the differential relationship dZ=dX-\theta(Z) dt+dL-dU, where X is a (0,\sigma) Brownian motion, and L and U are increasing processes that enforce a lower reflecting barrier at Z=0 and an upper reflecting barrier at Z=b, respectively. The cumulative cost process increases according to the differential relationship d\xi =c(\theta(Z)) dt+p dU, where c(\cdot) is a nondecreasing cost of control and p>0 is a penalty rate associated with displacement at the upper boundary. The objective is to minimize long-run average cost. This problem is solved explicitly, which allows one to also solve the following, essentially equivalent formulation: minimize the long-run average cost of control subject to an upper bound constraint on the average rate at which U increases. The two special problem features that allow an explicit solution are the use of a long-run average cost criterion, as opposed to a discounted cost criterion, and the lack of state-related costs other than boundary displacement penalties. The application of this theory to power control in wireless communication is discussed.Comment: Published at http://dx.doi.org/10.1214/105051604000000855 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Optimal Stopping Rules and Maximal Inequalities for Bessel Processes

We consider, for Bessel processes X ∈ Besα with arbitrary order (dimension) α ∈ R, the problem of the optimal stopping (1.4) for which the gain is determined by the value of the maximum of the process X and the cost which is proportional to the duration of the observation time. We give a description of the optimal stopping rule structure (Theorem 1) and the price (Theorem 2). These results are used for the proof of maximal inequalities of the type E max Xrr≤r ≤ γ(α) is a constant depending on the dimension (order) α. It is shown that γ(α) ∼ √α at α → ∞

On the Properties of a Tree-Structured Server Process

Let X0 be a nonnegative integer-valued random variable and let an independent copy of X0 be assigned to each leaf of a binary tree of depth k. If X0 and X0′ are adjacent leaves, let X1=(X0−1)++(X0′−1)+ be assigned to the parent node. In general, if Xj and Xj′ are assigned to adjacent nodes at level j = 0,⋯, k − 1, then Xj and Xj′ are, in turn, independent and the value assigned to their parent node is then Xj+1=(Xj−1)++(Xj′−1)+. We ask what is the behavior of Xk as k→∞. We give sufficient conditions for Xk→∞ and for Xk→0 and ask whether these are the only nontrivial possibilities. The problem is of interest because it asks for the asymptotics of a nonlinear transform which has an expansive term (the + in the sense of addition) and a contractive term (the + in the sense of positive part)

Stationary Gaussian Markov Processes as Limits of Stationary Autoregressive Time Series

We consider the class, ℂp, of all zero mean stationary Gaussian processes, {Yt : t ∈ (—∞, ∞)} with p derivatives, for which the vector valued process {(Yt(0) ,...,Yt(p)) : t ≥ 0} is a p + 1-vector Markov process, where Yt(0) = Y(t). We provide a rigorous description and treatment of these stationary Gaussian processes as limits of stationary AR(p) time series

First-Digit Law in Nonextensive Statistics

Nonextensive statistics, characterized by a nonextensive parameter $q$, is a promising and practically useful generalization of the Boltzmann statistics to describe power-law behaviors from physical and social observations. We here explore the unevenness of the first digit distribution of nonextensive statistics analytically and numerically. We find that the first-digit distribution follows Benford's law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature. The fluctuation decreases when $q$ increases, and the result converges to Benford's law exactly as $q$ approaches 2. The relevant regularities between nonextensive statistics and Benford's law are also presented and discussed.Comment: 11 pages, 3 figures, published in Phys. Rev.

Maximum-likelihood absorption tomography

Maximum-likelihood methods are applied to the problem of absorption tomography. The reconstruction is done with the help of an iterative algorithm. We show how the statistics of the illuminating beam can be incorporated into the reconstruction. The proposed reconstruction method can be considered as a useful alternative in the extreme cases where the standard ill-posed direct-inversion methods fail.Comment: 7 pages, 5 figure

Limit Distributions of Self-Normalized Sums

If Xi are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of Sn(2) =(∑ni=1 Xi)/(∑nj=1Xj2)1/2 as n→∞ has density f(t) = (2π)−1/2 exp(−t2/2) by the central limit theorem and the law of large numbers. If the tails of Xi are sufficiently smooth and satisfy P(Xi \u3e t) ∼ rt−α and P(Xi \u3c −t) ∼ lt−α as t→∞, where 0 \u3c α \u3c 2, r \u3e 0, l \u3e 0, Sn(2) still has a limiting distribution F even though Xi has infinite variance. The density f of F depends on α as well as on r/l. We also study the limiting distribution of the more general Sn(p) = (∑ni=1Xi)/(∑nj=1 |Xj|p)1/p where Xi are i.i.d. and in the domain of a stable law G with tails as above. In the cases p = 2 (see (4.21)) and p = 1 (see (3.7)) we obtain exact, computable formulas for f(t) = f(t,α,r/l), and give graphs of f for a number of values of α and r/l. For p = 2, we find that f is always symmetric about zero on (−1,1), even though f is symmetric on (−∞,∞) only when r = l

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like $|\hbox{\rm Im}\,z|$. We present the low and high disorder asymptotic behaviors. Then we particularize to the large $n$ limit of the average density of complex roots of monic algebraic polynomials of the form $P_n(z) = z^n +\sum_{k=1}^{n}c_kz^{n-k}$ with real independent, identically distributed Gaussian coefficients having zero mean and dispersion $\delta = \frac 1{\sqrt{n\lambda}}$. The average density tends to a simple, {\em universal} function of $\xi={2n}{\log |z|}$ and $\lambda$ in the domain $\xi\coth \frac{\xi}{2}\ll n|\sin \arg (z)|$ where nearly all the roots are located for large $n$.Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed tarfile (.66MB) containing 8 Postscript figures is available by e-mail from [email protected]

On leaders and condensates in a growing network

The Bianconi-Barabasi model of a growing network is revisited. This model, defined by a preferential attachment rule involving both the degrees of the nodes and their intrinsic fitnesses, has the fundamental property to undergo a phase transition to a condensed phase below some finite critical temperature, for an appropriate choice of the distribution of fitnesses. At high temperature it exhibits a crossover to the Barabasi-Albert model, and at low temperature, where the fitness landscape becomes very rugged, a crossover to the recently introduced record-driven growth process. We first present an analysis of the history of leaders, the leader being defined as the node with largest degree at a given time. In the generic finite-temperature regime, new leaders appear endlessly, albeit on a doubly logarithmic time scale, i.e., extremely slowly. We then give a novel picture for the dynamics in the condensed phase. The latter is characterized by an infinite hierarchy of condensates, whose sizes are non-self-averaging and keep fluctuating forever.Comment: 29 pages, 13 figures, 3 tables. A few minor change