Range-Renewal Processes: SLLNs and Power Laws*

Given n samples(viewed as an n-tuple) of a γ-regular discrete distribution π,in this article the authors concern with the weighted and unweighted graphs induced by the n samples. They first prove a series of SLLN results(of Dvoretzky-Erd¨os’ type). Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1 + γ. They also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distribu...     

Covering with blocks in the non-symmetric case

Considering an infinite string of i.i.d. random letters drawn from a finite alphabet we define the cover timeW _n as the number of random letters needed until each pattern of lenghtn appears at least once as a substring. Sharp weak and a.s. limit results onW _n are known in the symmetric case, i.e., when the random letters are uniformly distributed over the alphabet. In this paper we determine the limit distribution ofW _n in the nonsymmetric case asn. Generalizations in terms of point processes are also proved.Dedicated to Endre Csáki on his 60th birthday.     

A law of the iterated logarithm for processes with independent increments

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Rate conservation laws: A survey

We survey the rate conservation law, RCL for short, arising in queues and related stochastic models. RCL was recognized as one of the fundamental principles to get relationships between time and embedded averages such as the extended Little's formulaH=G, but we show that it has other applications. For example, RCL is one of the important techniques for deriving equilibrium equations for stochastic processes. It is shown that the various techniques, including Mecke's formula for a stationary random measure, can be formulated as RCL. For this purpose, we start with a new definition of the rate with respect to a random measure, and generalize RCL by using it. We further introduce the notion of quasi-expectation, which is a certain extension of the ordinary expectation, and derive RCL applicable to the sample average results. It means that the sample average formulas such asH=G can be obtained as the stationary RCL in the quasi-expectation framework. We also survey several extensions of RCL and discuss examples. Throughout the paper, we would like to emphasize how results can be easily obtained by using a simple principle, RCL.     

Exact results for the asymmetric simple exclusion process with a blockage

We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one tor<1. Exact finitevolume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium phase transition, corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state. A series expansion inr, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Padé approximants based on this series, which make specific assumptions about the nature of the singularity atr=1, match numerical data for the infinite system to 1 part in 10⁴.     

Diffusion on the torus for Hamiltonian maps

For a mapping of the torusT ² we propose a definition of the diffusion coefficientD suggested by the solution of the diffusion equation ofT ². The definition ofD, based on the limit of moments of the invariant measure, depends on the set where an initial uniform distribution is assigned. For the algebraic automorphism of the torus the limit is proved to exist and to have the same value for almost all initial sets in the subfamily of parallelograms. Numerical results show that it has the same value for arbitrary polygons and for arbitrary moments.     

The studies of the effect of photostimulated changes in thin carbon films

Photostimulated changes in thin carbon films deposited onto fused silica substrates by reactive magnetron sputtering and irradiated by the light of the first (1060nm) and the second (530nm) harmonics of a QW-switched Nd³⁺ AVG laser were investigated. The changes are supposed to be connected with the reordering of chemical bounds.     

Pathwise rate- stability for input-output processes

An input-output processZ = {Z(t), t 0} is said to be-rate stable ifZ(t) = o((t)) for some non-negative function(t). We prove that the processZ is -rate stable under weak conditions that include the assumption that input satisfies a linear burstiness condition and Z is asymptotically average stable. In many cases of interest, the conditions for-rate-stability can be verified from input data. For example, using input information, we establish-rate stability of the workload for multiserver queues, an ATM multiplexer, and-rate stability of queue-length processes for infinite server queues.     

A tandem of discrete-time queues with arrivals and departures at each stage

A discrete-time system of a tandem of queues with exogenous arrivals and departures at each stage is considered. A customer leaving queuek–1 departs the system with probability 1– ^[k] and continues to queuek with probability ^[k] . Exogenous arrivals to each stage are i.i.d. at each time slot. An approximate analysis of the occupancy and busy-period distributions of each stage based on a General Busy-period with batches and Memoryless (geometric) Idle period renewal Process (GBMIP) provides improved performance over two-state Markov approximations and gives exact results when there are no interstage departures.This research was supported in part by NSF grant NCR-8708282.     

The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot⁽¹⁶⁾ on the tail distribution ofX(t). Lévy's lower functions ofX(t) are characterized by an integral test.