Infinite Mean Research Articles

B-stability

A random variable Y is branching stable (B-stable) for a nonnegative integer-valued random variable J with E(J)>1 if Y *J ∿cY for some scalar c, where Y *J is the sum of J independent copies of Y. We explore some aspects of this notion of stability and show that, for any Y 0 with finite nonzero mean, if we define Y n+1=Y n *J /E(J) then the sequence Y n converges in law to a random variable Y ∞ that is B-stable for J. Also Y ∞ is the unique B-stable law with mean E(Y 0). We also present results relating to random variables Y 0 with zero means and infinite means. The notion of B-stability arose in a scheme for cataloguing a large network of computers.

Low-complexity MMSE turbo equalization: a possible solution for EDGE

This paper deals with a low complexity receiver scheme where equalization and channel decoding are jointly optimized in an iterative process. We derive the theoretical transfer function of the infinite length linear minimum mean square error (MMSE) equalizer with a priori information. A practical implementation is exposed which employs the fast Fourier transform (FFT) to compute the equalizer coefficients, resulting in a low-complexity receiver structure. The performance of the proposed scheme is investigated for the enhanced general packet radio service (EGPRS) radio link. Simulation results show that significant power gains may be achieved with only a few (3-4) iterations. These results demonstrate that MMSE turbo equalization is an attractive candidate for single-carrier broadband wireless transmissions in long delay-spread environments.

Long range dependence in network traffic and the closed loop behaviour of buffers under adaptive window control

We consider an Internet link carrying http-like traffic, i.e., transfers of finite volume files arriving at random time instants. These file transfers are controlled by an adaptive window protocol (AWP); an example of such a protocol is TCP. We provide analysis for the auto-covariance function of the AWP-controlled traffic into the link’s buffer; this traffic, in general, cannot be represented by an on–off process. The analysis establishes that, for TCP-controlled transfer of Pareto-distributed file sizes with infinite second moment, the traffic into the link buffer is long range-dependent (LRD). We also develop an analysis for obtaining the stationary distribution of the link buffer occupancy under an AWP-controlled transfer of files sampled from some distribution. For any AWP, the analysis provides us with the Laplace–Stieltjes transform (LST) of the distribution of the link buffer occupancy process in terms of the functions defining the AWP and the file size distribution. The analysis also provides a necessary and a sufficient condition for the finiteness of the mean link buffer content; these conditions again have explicit dependence on the AWP used and the file size distribution. This establishes the sensitivity of the buffer occupancy process to the file size distribution. Combining the results from the above analyses, we provide various examples in which the closed loop control of an AWP results in finite mean link buffer occupancy even though the file sizes are Pareto-distributed (with infinite second moment), and the traffic into the link buffer is long range-dependent (with Hurst parameters which would suggest an infinite mean queue occupancy under open loop analysis). We also study the effect of window reductions due to active queue management and find that window reductions lead to further lightening of the tail of buffer occupancy distribution. The significance of this work is three-fold: (i) by looking at the window evolution as a function of the amount of data served and not as a function of time, this work provides a new framework for analysing various processes related to the link buffer under AWP-controlled transfer of files with a general file size distribution; (ii) it indicates that the buffer behaviour in the Internet may not be as poor as predicted from an open loop analysis of a queue fed with LRD traffic; and (iii) it shows that the buffer behaviour (and hence the throughput performance for finite buffers) is sensitive to the distribution of file sizes.

Huisi's Priliminary Step in samadhi of the Lotus sutra

The samadhi of the Lotus Sutra was taught to Zhiyi by Huisi on Mount Dasu. At that time Zhiyi reached a certain state. But Huisi judged what Zhiyi realized was the preliminary step. Huisi's preliminary step was to preach in accordance with the various living beings based on the Meaning of the Course of Ease and Bliss in the Lotus Sutra. There are two ways. One is to preach infinite meanings in accordance with living beings, the samadhi with infinite meanings (_??__??__??__??__??_). The other is to manifest various figures in accordance with living beings, the samadhi that manifest the various figures universally (_??__??__??__??__??__??_). After the two samadhi that manifest infinite meanings are obtained, the samadhi of the Lotus Sutra that melds the infinite meanings into one is obtained. That is the perfect state at last.

Models of anomalous diffusion: the subdiffusive case

The paper discusses a model for anomalous diffusion processes. Their one-point probability density functions (p.d.f.) are exact solutions of fractional diffusion equations. The model reflects the asymptotic behavior of a jump (anomalous random walk) process with random jump sizes and random inter-jump time intervals with infinite means (and variances) which do not satisfy the Law of Large Numbers. In the case when these intervals have a fractional exponential p.d.f., the fractional Komogorov–Feller equation for the corresponding anomalous diffusion is provided and methods of finding its solutions are discussed. Finally, some statistical properties of solutions of the related Langevin equation are studied. The subdiffusive case is explored in detail. The emphasis is on a rigorous presentation which, however, would be accessible to the physical sciences audience.

Advection and dispersion in time and space

Previous work showed how moving particles that rest along their trajectory lead to time-nonlocal advection–dispersion equations. If the waiting times have infinite mean, the model equation contains a fractional time derivative of order between 0 and 1. In this article, we develop a new advection–dispersion equation with an additional fractional time derivative of order between 1 and 2. Solutions to the equation are obtained by subordination. The form of the time derivative is related to the probability distribution of particle waiting times and the subordinator is given as the first passage time density of the waiting time process which is computed explicitly.

Limit theorems for continuous-time random walks with infinite mean waiting times

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

Limit theorems for continuous-time random walks with infinite mean waiting times

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

A super-stable motion with infinite mean branching

A class of finite measure-valued càdlàg superprocesses X with Neveu's (1992) continuous-state branching mechanism is constructed. To this end, we start from certain supercritical ( α, d, β)-superprocesses X ( β) with symmetric α-stable motion and (1+ β)-branching and prove convergence on path space as β↓0. The log-Laplace equation related to X has the locally non-Lipschitz function ulog u as non-linear term (instead of u 1+ β in the case of X ( β) ). It can nevertheless be shown to be well-posed. X has infinite expectation, is immortal in all finite times, propagates mass instantaneously everywhere in space, and has locally countably infinite biodiversity.

A super-stable motion with infinite mean branching

A class of finite measure-valued càdlàg superprocesses X with Neveu's (1992) continuous-state branching mechanism is constructed. To this end, we start from certain supercritical (α,d,β)-superprocesses X(β) with symmetric α-stable motion and (1+β)-branching and prove convergence on path space as β↓0. The log-Laplace equation related to X has the locally non-Lipschitz function ulogu as non-linear term (instead of u1+β in the case of X(β)). It can nevertheless be shown to be well-posed. X has infinite expectation, is immortal in all finite times, propagates mass instantaneously everywhere in space, and has locally countably infinite biodiversity. Nous construisons une classe de processus de branchement à valeurs mesures finis qui sont une extension naturelle de processus de branchement à valeurs réelles positives étudiés par Neveu (1992). Pour arriver a ce résultat nous commençons avec des (α,d,β)-superprocessus X(β) qui correspondent aux systèmes des particules dans lesquels le déplacement de masse est décrit par une loi α-stable et le branchement par une loi (1+β)-stable, et nous prouvons la convergence dans l'espace des trajectoires càdlàg larsque β↓0. L'équation log-Laplace qui est associée au le processus limite X comparteun terme non lineaireen u logu, qui n'est pas lipschitzien. Nous pouvons neamains démontrer que cette équation est bien-posée. X est d'ésperance infinie, est immortel à temp fini, propage sa masse instantanement dans tout l'espace, et a une diversité biologique localement infinie.

On Transience Conditions for Markov Chains and Random Walks

We prove a new transience criterion for Markov chains on an arbitrary state space and give a corollary for real-valued chains. We show by example that in the case of a homogeneous random walk with infinite mean the proposed sufficient conditions are close to those necessary. We give a new proof of the well-known criterion for finiteness of the supremum of a random walk.

AN ASYMPTOTICALLY RISK-EFFICIENT SAMPLE ALLOCATION FOR ESTIMATING THE MEAN WAITING TIME IN THE 1 and 1 QUEUES

In this paper, we consider the problem of optimally allocating a sampling budget between the interarrival times and service times for the stationary queue and stationary queue with the goal of minimizing the mean squared error of one of four system performance measures. The natural estimator of the mean waiting time, obtained by substituting estimators for the arrival rate and service rate in the formula for the mean waiting time, has infinite mean squared error and cannot be used. Using an alternative estimator, we propose a two-stage sampling plan and show that this plan is asymptotically risk-efficient in the sense that the mean squared error of the resulting estimator converges to the minimum mean squared error as the sampling budget gets large. Asymptotic normality and other properties of the two-stage estimator are also proved. Simulation results for the system are given to support the theoretical results. **Supported by grant MDA904-00-1-0025 from NSA.

The distribution of particles in the plane dispersed by a simple 3-dimensional diffusion process.

Populations of particles dispersed in the 2-dimensional plane from a single point-source may be grouped as focus expansion patterns, with an exponentially decreasing density, and more diffuse patterns with thicker tails. Exponentially decreasing distributions are often modelled as the result of 2-dimensional diffusion processes acting to disperse the particles, while thick-tailed distributions tend to be modelled by purely descriptive distributions. Models based on the Cauchy distribution have been suggested, but these have not been related to diffusion modelling. However, the distribution of particles dispersed from a point source by a 3-dimensional Brownian motion that incorporates a constant drift, under the condition that the particle starts at a given height and is stopped when it reaches the xy plane (zero height) may be shown to result in both slim-tailed exponentially decreasing densities, and thick-tailed polynomially decreasing densities with infinite mean travel distance from the source, depending on parameter values. The drift in the third coordinate represents gravitation, while the drift in the first and second represents a (constant) wind. Conditions for the density having exponentially decreasing tails is derived in terms of gravitation and wind, with a special emphasis on applications to light-weighted particles such as fungal spores.

Critical Bellman–Harris branching processes with infinite variance allowing state-dependent immigration

The branching processes with state-dependent immigration are considered as alternating regenerative processes. The main purpose is to demonstrate some new “regenerative” methods. Critical Bellman–Harris branching processes with state-dependent immigration are investigated and new limit theorems are obtained in the case of an infinite offspring variance and possibly infinite mean of the immigrants.

A Commodity Price Process with a Unique Continuous Invariant Distribution Having Infinite Mean

We present a model of a storable commodity with a price process different from those usually associated with agricultural commodities or exhaustible resources. This model has positive probability of zero output. For consumption equal to zero, price is infinite. In any initial period with finite price, the path of expected prices, conditional on current information, approaches infinity at a rate determined by the cost of storage, and the variance also approaches infinity. Price is expected to fall with probability one in finite time. Successive price realizations are always correlated, storage remains strictly positive, and expected consumption exceeds a strictly positive bound. There is a continuous limiting invariant distribution of price, with infinite mean. K

Dwell time modeling for wireless communication systems and problems of the velocity distribution

AbstractThe distribution of the dwell time, which represents a higher level of abstraction than the use of mobility models, is often required for the performance evaluation of mobile communication systems. This contribution investigates the problem of the dwell time modeling. Using the often employed model for a constant movement, it is shown that a suitable choice of the velocity distribution is very important. Thus, a condition for the probability density function (PDF) of the velocity is derived, which prevents an unbounded mean dwell time. Since an infinite mean dwell time causes some problems for me estimation of the mobility‐related signaling traffic, it is important that the mean dwell time is finite. Furthermore, some basic models for the PDF of the travel length of a subscriber within a radio cell are introduced and the relation wim realistic mobility patterns are shown.

Limit theorems for alternating renewal processes in the infinite mean case

The asymptotic behaviour of an occupation-time process associated with alternating renewal processes is investigated in the infinite mean cycle case. The limit theorems obtained extend some asymptotic results proved by Dynkin (1955), Lamperti (1958) and Erickson (1970) for the classical spent lifetime process. Some new phenomena are also presented.

Limit theorems for alternating renewal processes in the infinite mean case

The asymptotic behaviour of an occupation-time process associated with alternating renewal processes is investigated in the infinite mean cycle case. The limit theorems obtained extend some asymptotic results proved by Dynkin (1955), Lamperti (1958) and Erickson (1970) for the classical spent lifetime process. Some new phenomena are also presented.

Asymptotic properties and absolute continuity of laws stable by random weighted mean

We study properties of stable-like laws, which are solutions of the distributional equation Z = d ∑ i=1 NA iZ i, where ( N, A 1, A 2,…) is a given random variable with values in {0,1,…}×[0,∞)×[0,∞)×…, and Z, Z 1, Z 2,… are identically distributed positive random variables, independent of each other and independent of ( N, A 1, A 2,…). Examples of such laws contain the laws of the well-known limit random variables in: (a) the Galton–Watson process or general branching processes, (b) branching random walks, (c) multiplicative processes, and (d) smoothing processes. For any solution Z (with finite or infinite mean), we find asymptotic properties of the distribution function P( Z⩽ x) and those of the characteristic function Ee i tZ ; we prove that the distribution of Z is absolutely continuous on (0,∞), and that its support is the whole half-line [0,∞). Solutions which are not necessarily positive are also considered.

Regenerative processes in the infinite mean cycle case

A class of non-negative alternating regenerative processes is considered, where the process stays at zero random time (waiting period), then it jumps to a random positive level and hits zero after some random period (life period), depending on the evolution of the process. It is assumed that the waiting time and the lifetime belong to the domain of attraction of stable laws with parameters in the interval (½,1]. An integral representation for the distribution functions of the regenerative process is obtained, using the spent time distributions of the corresponding alternating renewal process. Given the asymptotic behaviour of the process in the regenerative cycle, different types of limiting distributions are proved, applying some new results for the corresponding renewal process and two limit theorems for the convergence in distribution.