Mittag-Leffler Distribution Research Articles

Renewal Theory for a System with Internal States

We investigate a stochastic signal described by a renewal process for a system with N states. Each state has an associated joint distribution for the signal’s intensity and its holding time. We calculate multi-point distributions, correlation functions, and the power-spectrum of the signal. Focusing on fat tailed power-law distributed sojourn times in the states of the system, we investigate 1/f noise in this widely applicable model. When the mean waiting time is infinite, the averaged sample spectrum depends both on the age of the process, i.e. the time elapsing from start of the process and the start of observation, and on the total time of observation. Fluctuations of the periodogram estimator of the power-spectrum are investigated for aged systems and are found to be determined by the distribution of the number of renewals in the observation time window. These reduce to the Mittag-Leffler distribution when the start of observation is also the start of the process. When the average waiting time is finite we find a time independent Wienerian spectrum computed from the stationary correlation function of the signal.

On Mittag-Leffler distributions and related stochastic processes

Random variables with Mittag-Leffler distribution can take values either in the set of non-negative integers or in the positive real line. They can be of two different types, one (type-1) heavy-tailed with index α∈(0,1), the other (type-2) possessing all its moments. We investigate various stochastic processes where they play a key role, among which: the discrete space/time Neveu branching process, the discrete-space continuous-time Neveu branching process, the continuous space/time Neveu branching process (CSBP) and renewal processes with rare events. Its relation to (discrete or continuous) self-decomposability and branching processes with immigration is emphasized. Special attention will be paid to the Neveu CSBP for its connection with the Bolthausen–Sznitman coalescent. In this context, and following a recent work of Möhle (2015), a type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu’s CSBP block-counting process arising in sampling from PD(e−t,0). Further combinatorial developments of this model are investigated.

Solvable non-Markovian dynamic network

Non-Markovian processes are widespread in natural and human-made systems, yet explicit modeling and analysis of such systems is underdeveloped. We consider a non-Markovian dynamic network with random link activation and deletion (RLAD) and heavy-tailed Mittag-Leffler distribution for the interevent times. We derive an analytically and computationally tractable system of Kolmogorov-like forward equations utilizing the Caputo derivative for the probability of having a given number of active links in the network and solve them. Simulations for the RLAD are also studied for power-law interevent times and we show excellent agreement with the Mittag-Leffler model. This agreement holds even when the RLAD network dynamics is coupled with the susceptible-infected-susceptible spreading dynamics. Thus, the analytically solvable Mittag-Leffler model provides an excellent approximation to the case when the network dynamics is characterized by power-law-distributed interevent times. We further discuss possible generalizations of our result.

Fractional derivative anomalous diffusion equation modeling prime number distribution

Abstract This study suggests that the power law decay of prime number distribution can be considered a sub-diffusion process, one of typical anomalous diffusions, and could be described by the fractional derivative equation model, whose solution is the statistical density function of Mittag-Leffler distribution. It is observed that the Mittag-Leffler distribution of the fractional derivative diffusion equation agrees well with the prime number distribution and performs far better than the prime number theory. Compared with the Riemann’s method, the fractional diffusion model is less accurate but has clear physical significance and appears more stable. We also find that the Shannon entropies of the Riemann’s description and the fractional diffusion models are both very close to the original entropy of prime numbers. The proposed model appears an attractive physical description of the power law decay of prime number distribution and opens a new methodology in this field.

Time series models associated with Mittag-Leffler type distributions and its properties

ABSTRACTWe have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order α(0 < α ⩽ 1). A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of this new model and its relevance to time series. Distribution of gliding sums, regression behaviors, and sample path properties are studied. Finally we introduce the q-Mittag-Leffler process associated with the q-Mittag-Leffler distribution.

Anomalous diffusion in a quenched-trap model on fractal lattices

Models with mixed origins of anomalous subdiffusion have been considered important for understanding transport in biological systems. Here one such mixed model, the quenched-trap model (QTM) on fractal lattices, is investigated. It is shown that both ensemble- and time-averaged mean-square displacements (MSDs) show subdiffusion with different scaling exponents, i.e., this system shows weak ergodicity breaking. Moreover, time-averaged MSD exhibits aging and converges to a random variable following the modified Mittag-Leffler distribution. It is also shown that the QTM on a fractal lattice cannot be reduced to the continuous-time random walks if the spectral dimension of the fractal lattice is less than 2.

A relative entropy method to measure non-exponential random data

Abstract This paper develops a relative entropy method to measure non-exponential random data in conjunction with fractional order moment, logarithmic moment and tail statistics of Mittag–Leffler distribution. The distribution of non-exponential random data follows neither the exponential distribution nor exponential decay. The proposed strategy is validated by analyzing the experiment data, which are generated by Monte Carlo method using Mittag–Leffler distribution. Compared with the traditional Shannon entropy, the relative entropy method is simple to be implemented, and its corresponding relative entropies approximated by the fractional order moment, logarithmic moment and tail statistics can easily and accurately detect the non-exponential random data.

Distributional Behavior of Time Averages of Non- $$L^1$$ L 1 Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of \(L^1(m)\) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-\(L^1(m)\) functions. Here, we provide another distributional behavior of time averages of non-\(L^1(m)\) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-\(L^1(m)\) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-\(L^1(m)\) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-\(L^1(m)\) function in the one-dimensional intermittent maps.

Fractional Skellam processes with applications to finance

Abstract The recent literature on high frequency financial data includes models that use the difference of two Poisson processes, and incorporate a Skellam distribution for forward prices. The exponential distribution of inter-arrival times in these models is not always supported by data. Fractional generalization of Poisson process, or fractional Poisson process, overcomes this limitation and has Mittag-Leffler distribution of inter-arrival times. This paper defines fractional Skellam processes via the time changes in Poisson and Skellam processes by an inverse of a standard stable subordinator. An application to high frequency financial data set is provided to illustrate the advantages of models based on fractional Skellam processes.

Bridge fatigue life prediction using Mittag–Leffler distribution

ABSTRACTThis paper is aimed at developing a statistical approach for fatigue life prediction of steel bridges with structural health monitoring data. The present approach obtains the static cumulative damage index by using the Mittag–Leffler distribution in conjunction with the statistical form of Miner's rule and investigates the influence of annual traffic increase on the evolution of the dynamic service life by a modified statistical formula of Miner's rule, which includes the relationship between the levels of stress range and the annual traffic increase rate. The proposed method is exemplified to specifically predict the static fatigue life of the Neville Island Bridge and the dynamic fatigue life of the Birmingham Bridge.

Aging generates regular motions in weakly chaotic systems

Using intermittent maps with infinite invariant measures, we investigate the universality of time-averaged observables under aging conditions. According to Aaronson's Darling-Kac theorem, in non-aged dynamical systems with infinite invariant measures, the distribution of the normalized time averages of integrable functions converges to the Mittag-Leffler distribution. This well known theorem holds when the start of observations coincides with the start of the dynamical processes. Introducing a concept of the aging limit where the aging time ${t}_{a}$ and the total measurement time $t$ goes to infinity while the aging ratio ${t}_{a}/t$ is kept constant, we obtain a novel distributional limit theorem of time-averaged observables integrable with respect to the infinite invariant density. Applying the theorem to the Lyapunov exponent in intermittent maps, we find that regular motions and a weakly chaotic behavior coexist in the aging limit. This mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in Hamiltonian systems. The probability of finding regular motions in non-aged processes is zero, while in the aging regime it is finite and it increases when system ages.

Ergodic properties of continuous-time random walks: Finite-size effects and ensemble dependences

The effects of spatial confinements and smooth cutoffs of the waiting time distribution in continuous-time random walks (CTRWs) are studied analytically. We also investigate dependences of ergodic properties on initial ensembles (i.e., distributions of the first waiting time). Here, we consider two ensembles: the equilibrium and a typical non-equilibrium ensembles. For both ensembles, it is shown that the time-averaged mean square displacement (TAMSD) exhibits a crossover from normal to anomalous diffusion due to the spacial confinement and this crossover does not vanish even in the long measurement time limit. Moreover, for the non-equilibrium ensemble, we show that the probability density function of the diffusion constant of TAMSD follows the transient Mittag-Leffler distribution, and that scatter in the TAMSD shows a clear transition from weak ergodicity breaking (an irreproducible regime) to ordinary ergodic behavior (a reproducible regime) as the measurement time increases. This convergence to ordinary ergodicity requires a long measurement time compared to common distributions such as the exponential distribution; in other words, the weak ergodicity breaking persists for a long time. In addition, it is shown that, besides the TAMSD, a class of observables also exhibits this slow convergence to ergodicity. We also point out that, even though the system with the equilibrium initial ensemble shows no aging, its behavior is quite similar to that for the non-equilibrium ensemble.

Limiting distribution of the maximum of a null recurrent diffusion process

A limit theorem for the maximum processes of a class of null recurrent linear diffusions is proved. The limiting distribution is a mixture of the Mittag-Leffler distribution.

INTEGER VALUED AUTOREGRESSIVE PROCESSES WITH GENERALIZED DISCRETE MITTAG-LEFFLER MARGINALS

In this paper we consider a generalization of discrete Mittag-Leffler distributions. We introduce and study the properties of a new distribution called geometric generalized discrete Mittag-Leffler distribution. Autoregressive processes with geometric generalized discrete Mittag-Leffler distributions are developed and studied. The distributions are further extended to develop a more general class of geometric generalized discrete semi-Mittag-Leffler distributions. The processes are extended to higher orders also. An application with respect to an empirical data on customer arrivals in a bank counter is also given. Various areas of potential applications like human resource development, insect growth, epidemic modeling, industrial risk modeling, insurance and actuaries, town planning etc are also discussed.

Intrinsic randomness of transport coefficient in subdiffusion with static disorder

Fluctuations in the time-averaged mean-square displacement for random walks on hypercubic lattices with static disorder are investigated. It is analytically shown that the diffusion coefficient becomes a random variable as a manifestation of weak ergodicity breaking. For two- and higher- dimensional systems, the distribution function of the diffusion coefficient is found to be the Mittag-Leffler distribution, which is the same as for the continuous-time random walk, whereas for one-dimensional systems a different distribution (a modified Mittag-Leffler distribution) arises. We also present a comparison of these two distributions in terms of an ergodicity-breaking parameter and show that the modified Mittag-Leffler distribution has a larger deviation from ergodicity. Some remarks on similarities between these results and observations in biological experiments are presented.

Subordinated diffusion and continuous time random walk asymptotics

Anomalous transport is usually described either by models of continuous time random walks (CTRWs) or, otherwise, by fractional Fokker-Planck equations (FFPEs). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a Lévy α-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates a trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit.

Role of infinite invariant measure in deterministic subdiffusion

Statistical properties of the transport coefficient for deterministic subdiffusion are investigated from the viewpoint of infinite ergodic theory. We find that the averaged diffusion coefficient is characterized by the infinite invariant measure of the reduced map. We also show that when the time difference is much smaller than the total observation time, the time-averaged mean square displacement depends linearly on the time difference. Furthermore, the diffusion coefficient becomes a random variable and its limit distribution is characterized by the universal law called the Mittag-Leffler distribution.

On the Local Time of Random Walks Associated with Gegenbauer Polynomials

The local time of random walks associated with Gegenbauer polynomials \(P_{n}^{(\alpha)}(x)\), x∈[−1,1], is studied in the recurrent case: \(\alpha\in [-\frac{1}{2},0]\). When α is nonzero, the limit distribution is given in terms of a Mittag-Leffler distribution. The proof is based on a local limit theorem for the random walk associated with Gegenbauer polynomials. As a by-product, we derive the limit distribution of the local time of some particular birth-and-death Markov chains on ℕ.

A Generalization to Bivariate Mittag–Leffler and Bivariate Discrete Mittag–Leffler Autoregressive Processes

A generalization of the Gaver and Lewis (1980) model of first-order autoregressive process with marginals as bivariate Mittag–Leffler distribution is obtained. A necessary and sufficient condition for stationarity of the process is established. Autoregressive process with marginals follow bivariate discrete Mittag–Leffler distribution is also developed. The unknown parameters of the processes are estimated and some numerical results of the estimations are given.

Stochastic calculus for uncoupled continuous-time random walks

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications not only in physics but also in insurance, finance, and economics. A definition is given for a class of stochastic integrals driven by a CTRW, which includes the Itō and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is a martingale. It is proved that, as a consequence of the martingale transform theorem, if the CTRW is a martingale, the Itō integral is a martingale too. It is shown how the definition of the stochastic integrals can be used to easily compute them by Monte Carlo simulation. The relations between a CTRW, its quadratic variation, its Stratonovich integral, and its Itō integral are highlighted by numerical calculations when the jumps in space of the CTRW have a symmetric Lévy alpha -stable distribution and its waiting times have a one-parameter Mittag-Leffler distribution. Remarkably, these distributions have fat tails and an unbounded quadratic variation. In the diffusive limit of vanishing scale parameters, the probability density of this kind of CTRW satisfies the space-time fractional diffusion equation (FDE) or more in general the fractional Fokker-Planck equation, which generalizes the standard diffusion equation, solved by the probability density of the Wiener process, and thus provides a phenomenologic model of anomalous diffusion. We also provide an analytic expression for the quadratic variation of the stochastic process described by the FDE and check it by Monte Carlo.