Showing papers on "Renewal theory published in 2021"

On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics

Abstract: Recently the so-called Prabhakar generalization of the fractional Poisson counting process attracted much interest for his flexibility to adapt to real world situations. In this renewal process the waiting times between events are IID continuous random variables. In the present paper we analyze discrete-time counterparts: Renewal processes with integer IID interarrival times which converge in well-scaled continuous-time limits to the Prabhakar-generalized fractional Poisson process. These processes exhibit non-Markovian features and long-time memory effects. We recover for special choices of parameters the discrete-time versions of classical cases, such as the fractional Bernoulli process and the standard Bernoulli process as discrete-time approximations of the fractional Poisson and the standard Poisson process, respectively. We derive difference equations of generalized fractional type that govern these discrete time-processes where in well-scaled continuous-time limits known evolution equations of generalized fractional Prabhakar type are recovered. We also develop in Montroll–Weiss fashion the “Prabhakar Discrete-time random walk (DTRW)” as a random walk on a graph time-changed with a discrete-time version of Prabhakar renewal process. We derive the generalized fractional discrete-time Kolmogorov–Feller difference equations governing the resulting stochastic motion. Prabhakar-discrete-time processes open a promising field capturing several aspects in the dynamics of complex systems.

Maximum of Catalytic Branching Random Walk with Regularly Varying Tails

Abstract: For a continuous-time catalytic branching random walk (CBRW) on $${\mathbb {Z}}$$ , with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include regular variation of the jump distribution tail for underlying random walk and the well-known $$L\log L$$ condition for the offspring numbers. In our classification, given in Bulinskaya (Theory Probab Appl 59(4):545–566, 2015), the analysis refers to supercritical CBRW. The principal result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization, and nonlinear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with “heavy” tails, in contrast to the known behavior in case of “light” tails of the random walk jumps. The new tools such as “many-to-few lemma” and spinal decomposition appear ineffective here. The approach developed in this paper combines the techniques of renewal theory, Laplace transform, nonlinear integral equations and large deviations theory for random sums of random variables.

Sharp polynomial bounds on decay of correlations for multidimensional nonuniformly hyperbolic systems and billiards

Abstract: Gouezel and Sarig introduced operator renewal theory as a method to prove sharp results on polynomial decay of correlations for certain classes of nonuniformly expanding maps. In this paper, we apply the method to planar dispersing billiards and multidimensional nonMarkovian intermittent maps.

Point and Interval Estimators of Reliability Indices for Repairable Systems Using the Weibull Generalized Renewal Process

Abstract: The generalized renewal process considers repair efficiency of imperfect repair in reliability assessment of repairable systems; therefore, its evaluation results are close to real repair circumstance than the ordinary renewal process or the non-homogeneous Poisson process. Based on the asymptotic distribution of maximum likelihood estimation, a calculating method of reliability indices for repairable systems with imperfect repair is proposed. The point and approximate interval estimators of model parameters for Kijima type Weibull generalized renewal processes Models I and II, as well as reliability indices of repairable systems, such as reliability, cumulative failure number and failure intensity, etc., are all presented. Two real cases are studied using generalized renewal processes Models I and II respectively to show the validity of our method. The results show that imperfect repair makes the instantaneous failure intensity of a repairable system discretely jump either up or down at the time of each failure, and the method proposed in this paper agrees well with the other exiting methods, and can also reduce the complexity of calculation.

Uncertain insurance risk process with single premium and multiple classes of claims

Abstract: Traditionally an insurance risk process is considered under the framework of probability theory with a prerequisite that the estimated distribution function is close enough to the real frequency. However, due to economic or technological reasons, sometimes data are unavailable or difficult to obtain, such as when we consider a new insurance product or insurance for valuable weapons. Under these situations, reimbursement policies are based on experts’ belief degree, which has a much wider range than the real frequency. As a result, we should employ uncertain insurance risk models to better deal with human uncertainty in running an insurance company. Noticing the fact that an insurance company pays for different kinds of risks and the uncertainty of the customer’s arrivals and payments, we investigate an uncertain insurance risk process with multiple classes of claims where the premium process follows an uncertain renewal process. Then we derive expressions for the ruin index and the uncertainty distribution of the ruin time. Some numerical examples and a real data example are performed to capture more insights.

Nonparametric Bayesian Modeling and Estimation for Renewal Processes

Abstract: We propose a flexible approach to modeling for renewal processes. The model is built from a structured mixture of Erlang densities for the renewal process inter-arrival density. The Erlang mixture ...

Reconceptualising Atrial Fibrillation Using Renewal Theory: A Novel Approach to the Assessment of Atrial Fibrillation Dynamics.

Abstract: Despite a century of research, the mechanisms of AF remain unresolved. A universal motif within AF research has been unstable re-entry, but this remains poorly characterised, with competing key conceptual paradigms of multiple wavelets and more driving rotors. Understanding the mechanisms of AF is clinically relevant, especially with regard to treatment and ablation of the more persistent forms of AF. Here, the authors outline the surprising but reproducible finding that unstable re-entrant circuits are born and destroyed at quasi-stationary rates, a finding based on a branch of mathematics known as renewal theory. Renewal theory may be a way to potentially unify the multiple wavelet and rotor theories. The renewal rate constants are potentially attractive because they are temporally stable parameters of a defined probability distribution (the exponential distribution) and can be estimated with precision and accuracy due to the principles of renewal theory. In this perspective review, this new representational architecture for AF is explained and placed into context, and the clinical and mechanistic implications are discussed.

Analysis of Strategic Market Management in Light of Stochastic Processes, Recurrence Relation, Abelian Group and Expectation

Abstract: This paper entails a novel approach of analysis of strategic market management based on renewal reward stochastic process. The paper has also pointed out a discovered fact that clearly infers realization of cost analysis of product revalidation in light of Brownian motion with drift. The paper indicates a rare and new concept of how can compound stochastic process be applied to sense business cost analysis. In demand–supply analysis, there lies the essence of realization of alternating renewal theory-based customer satisfaction. Furthermore, the paper also shows a novel analysis of product upgradation in light of conditional expectation and simple random walk. Facts related to recurrence relation, Abelian group and expectation indicate a non-conventional approach of business gain prediction.

Markov chains with exponential return times are finitary

Abstract: Consider an ergodic Markov chain on a countable state space for which the return times have exponential tails. We show that the stationary version of any such chain is a finitary factor of an i.i.d. process. A key step is to show that any stationary renewal process whose jump distribution has exponential tails and is not supported on a proper subgroup of $\mathbb{Z}$ is a finitary factor of an i.i.d. process.

Close latency-security trade-off for the Nakamoto consensus

Abstract: Bitcoin is a peer-to-peer electronic cash system invented by Nakamoto in 2008. While it has attracted much research interest, its exact latency and security properties remain open. Existing analyses provide security and latency (or confirmation time) guarantees that are too loose for practical use. In fact the best known upper bounds are several orders of magnitude larger than a lower bound due to a well-known private-mining attack. This paper describes a continuous-time model for blockchains and develops a rigorous analysis that yields close upper and lower bounds for the latency-security trade-off. For example, when the adversary controls 10% of the total mining power and the block propagation delays are within 10 seconds, a Bitcoin block is secured with less than 10-3 error probability if it is confirmed after four hours, or with less than 10-9 error probability if confirmed after ten hours. These confirmation times are about two hours away from their corresponding lower bounds. To establish such close bounds, the blockchain security question is reduced to a race between the Poisson adversarial mining process and a renewal process formed by a certain species of honest blocks. The moment generation functions of relevant renewal times are derived in closed form. The general formulas from the analysis are then applied to study the latency-security trade-off of several well-known proof-of-work longest-chain cryptocurrencies. Guidance is also provided on how to set parameters for different purposes.

Estimating the inter-occurrence time distribution from superposed renewal processes

Abstract: Superposition of renewal processes is common in practice, and it is challenging to estimate the distribution of the individual inter-occurrence time associated with the renewal process. This is because with only aggregated event history, the link between the observed recurrence times and the respective renewal processes are completely missing, rendering existing theory and methods inapplicable. In this article, we propose a nonparametric procedure to estimate the inter-occurrence time distribution by properly deconvoluting the renewal equation with the empirical renewal function. By carefully controlling the discretization errors and properly handling challenges due to implicit and non-smooth mapping via the renewal equation, our theoretical analysis establishes the consistency and asymptotic normality of the nonparametric estimators. The proposed nonparametric distribution estimators are then utilized for developing theoretically valid and computationally efficient inferences when a parametric family is assumed for the individual renewal process. Comprehensive simulations show that compared with the existing maximum likelihood method, the proposed parametric estimation procedure is much faster, and the proposed estimators are more robust to round-off errors in the observed data.

Limit theorems for hawkes processes including inhibition

Abstract: In this paper we consider some non linear Hawkes processes with signed reproduction function (or memory kernel) thus exhibiting both self-excitation and inhibition. We provide a Law of Large Numbers, a Central Limit Theorem and large deviation results, as time growths to infinity. The proofs lie on a renewal structure for these processes introduced in Costa-Graham-Marsalle-Tran (2020) which leads to a comparison with cumulative processes. Explicit computations are made on some examples. Similar results have been obtained in the literature for self-exciting Hawkes processes only.

Exact Results and Bounds for the Joint Tail and Moments of the Recurrence Times in a Renewal Process

Abstract: The best known result about the joint distribution of the backward and forward recurrence times in a renewal process concerns the asymptotic behavior for the tail of that bivariate distribution. In the present paper we study the joint behavior of the recurrence times at a fixed time point t, and we obtain both exact results and bounds for their joint tail behavior. We also obtain results about the joint moments of these two random variables and we show in particular that the expectation of the product between the two recurrence times increases with time when the interarrival distribution has a decreasing failure rate. The results are illustrated by some numerical examples.

Log-concavity of compound distributions with applications in operational and actuarial models

Abstract: We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model. We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

Monte Carlo Simulation

Abstract: This chapter discusses the basic concept and techniques for Monte Carlo simulation. The simulation methods for a single random variable as well as those for a random vector (consisting of multiple variables) are discussed, followed by the simulation of some special stochastic processes, including Poisson process, renewal process, Gamma process and Markov process. Some advanced simulation techniques, such as the importance sampling, Latin hypercube sampling, and subset simulation, are also addressed in this chapter.

Critical fluctuations in renewal models of statistical mechanics

Abstract: We investigate the sharp asymptotic behavior at criticality of the large fluctuations of extensive observables in renewal models of statistical mechanics, such as the Poland–Scheraga model of DNA denaturation, the Fisher–Felderhof model of fluids, the Wako–Saito–Munoz–Eaton model of protein folding, and the Tokar–Dreysse model of strained epitaxy. These models amount to Gibbs changes of measure of a classical renewal process and can be identified with a constrained pinning model of polymers. The extensive observables that enter the thermodynamic description turn out to be cumulative rewards corresponding to deterministic rewards that are uniquely determined by the waiting time and grow no faster than it. The probability decay with the system size of their fluctuations switches from exponential to subexponential at criticality, which is a regime corresponding to a discontinuous pinning–depinning phase transition. We describe such decay by proposing a precise large deviation principle under the assumption that the subexponential correction term to the waiting time distribution is regularly varying. This principle is, in particular, used to characterize the fluctuations of the number of renewals, which measures the DNA-bound monomers in the Poland–Scheraga model, the particles in the Fisher–Felderhof model and the Tokar–Dreysse model, and the native peptide bonds in the Wako–Saito–Munoz–Eaton model.

Ruin and Dividend Measures in the Renewal Dual Risk Model

Abstract: In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under study have been the calculation of expected discounted future dividends and optimal strategies, where the dividend calculation have been done on aggregate. We can find works, at first using the classical compound Poisson model, then some examples of other renewal Erlang models. Knowing that ruin is ultimately achieved, we find important that dividends should be evaluated on an individual basis, where the early dividend contribution for the aggregate are of utmost importance. From our calculations we can really see how much important is the contribution of the first dividend. Afonso et al. (Insur Math Econ, 53(3), 906–918, 2013) had worked similar problems for the classical compound Poisson dual model. Besides that we find explicit formulae for both the probability of getting a dividend and the distribution of the amount of a single dividend. We still work the probability distribution of the number of gains to reach a given upper target (like a constant dividend barrier) as well as for the number of gains down to ruin. We complete the study working some illustrative numerical examples that show final numbers for the several problems under study.

On ruin probabilities with risky investments in a stock with stochastic volatility

Abstract: We investigate the asymptotic of ruin probabilities when the company combines the life- and non-life insurance businesses and invests its reserve into a risky asset with stochastic volatility and drift driven by a two-state Markov process. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities.

Bounds for mixing times for finite semi-Markov processes with heavy-tail jump distribution

Abstract: Consider a Markov chain with nite state space and suppose you wish to change time replacing the integer step index $n$ with a random counting process $N(t)$. What happens to the mixing time of the Markov chain? We present a partial reply in a particular case of interest in which $N(t)$ is a counting renewal process with power-law distributed inter-arrival times of index $\beta$. We then focus on $\beta \in (0,1)$ leading to infinite expectation for inter-arrival times and further study the situation in which inter-arrival times follow the Mittag-Leffler distribution of order $\beta$.

Stability Analysis of a Multi-class Retrial Queue with General Retrials and Classical Retrial Policy

Abstract: We deal with a single server multi-class retrial model, feed by Poisson input. The system is considered under classical retrial policy, while inter-retrial times are class dependent and generally distributed. Such systems have various applications like multi-access protocols or cellular mobile networks, where blocked messages are sent again after some waiting period. We rely on regenerative approach and results from renewal theory to obtain the stability criterion of the system under consideration and present some simulation results, to illustrate that obtained condition could be extended to the case with general input.

Synchronized abandonment in discrete-time renewal input queues with vacations

Abstract: We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.

Optimal harvesting strategy based on uncertain logistic population model

Abstract: The variations of population size with respect to time are often described by means of differential equations. This paper assumes the population size follows an uncertain logistic population equation, and calculates its uncertainty distribution and α -paths. The first hitting time that the population size reaches a pre-set level is investigated, which forms an uncertain renewal process, based on which a harvesting strategy is designed. With the help of fundamental theorem of uncertain renewal processes, the optimal harvesting strategy problem is transformed to a traditional optimization problem involving two variables which could be easily solved analytically or numerically.

Damped oscillations of the probability of random events followed by absolute refractory period: exact analytical results

Abstract: There are numerous examples of natural and artificial processes that represent stochastic sequences of events followed by an absolute refractory period during which the occurrence of a subsequent event is impossible. In the simplest case of a generalized Bernoulli scheme for uniform random events followed by the absolute refractory period, the event probability as a function of time can exhibit damped transient oscillations. Using stochastically-spiking point neuron as a model example, we present an exact and compact analytical description for the oscillations without invoking the standard renewal theory. The resulting formulas stand out for their relative simplicity, allowing one to analytically obtain the amplitude damping of the 2nd and 3rd peaks of the event probability.