Limiting distributions of continuous-time random walks with superheavy-tailed waiting times

We study the long-time behavior of the scaled walker (particle) position associated with decoupled continuous-time random walk which is characterized by superheavy-tailed distribution of waiting times and asymmetric heavy-tailed distribution of jump lengths. Both the scaling function and the corresponding limiting probability density are determined for all admissible values of tail indexes describing the jump distribution. To analytically investigate the limiting density function, we derive a number of different representations of this function and, by this way, establish its main properties. We also develop an efficient numerical method for computing the limiting probability density and compare our analytical and numerical results.Comment: 35 pages, 4 figure

Big jump principle for heavy-tailed random walks with correlated increments

The big jump principle explains the emergence of extreme events for physical quantities modelled by a sum of independent and identically distributed random variables which are heavy-tailed. Extreme events are large values of the sum and they are solely dominated by the largest summand called the big jump. Recently, the principle was introduced into physical sciences where systems usually exhibit correlations. Here, we study the principle for a random walk with correlated increments. Examples are the autoregressive model of first order and the discretized Ornstein-Uhlenbeck process both with heavy-tailed noise. The correlation leads to the dependence of large values of the sum not only on the big jump but also on the following increments. We describe this behaviour by two big jump principles, namely unconditioned and conditioned on the step number when the big jump occurs. The unconditional big jump principle is described by a correlation dependent shift between the sum and maximum distribution tails. For the conditional big jump principle, the shift depends also on the step number of the big jump

Some non-standard approaches to the study of sums of heavy-tailed distributions

Heavy-tailed phenomena arise whenever events with very low probability have sufficiently large consequences that these events cannot be treated as negligible. These are sometimes described as low intensity, high impact events. Sums of heavy-tailed random variables play a major role in many areas of applied probability, for instance in risk theory, insurance mathematics, financial mathematics, queueing theory, telecommunications and computing, to name but a few areas. The theory of the asymptotic behaviour of a sum of independent heavy-tailed random variables is well-understood. We give a review of known results in this area, stressing the importance of some insensitivity properties of the class of long-tailed distributions. We introduce the new concept of the Boundary Class for a long-tailed distribution, and describe some of its properties and uses. We give examples of calculating the boundary class. Geometric sums of random variables are a useful model in their own right, for instance in reliability theory, but are also useful because they model the maximum of a random walk, which is itself a model that occurs in many applications. When the summands are heavy-tailed and independent then the asymptotic behaviour has been known since the 1970s. The asymptotic expression for the geometric sum is often used as an approximation to the actual distribution, owing to the (usually) analytically intractable form of the exact distribution. However the accuracy of this asymptotic approximation can be very poor, as we demonstrate. Following and further developing work by Kalashnikov and Tsitsiashvili we construct an upper bound for the relative accuracy of this approximation. We then develop new techniques for the application of our analytical results, and apply these in practice to several examples. Source code viii for the computer algorithms used in these calculations is given. As we have said, the asymptotic behaviour of a sum of heavy-tailed random variables is well-understood when the random variables are independent, the main characteristic being the principle of the single big jump. However, the case when the random variables are dependent is much less clear. We study this case for both deterministic and random sums using a novel approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables still to hold. We give several examples to show how to apply and check our conditions, and the examples demonstrate a variety of effects owing to the dependence, and are also interesting in their own right. All the results we develop on this topic are entirely new. Some of the examples also include results that are new and have not been obtainable through previously existing techniques. For some examples we study the asymptotic behaviour is known, and this allows us to contrast our approach with previous approaches

Modeling record-breaking stock prices

We study the statistics of record-breaking events in daily stock prices of 366 stocks from the Standard and Poors 500 stock index. Both the record events in the daily stock prices themselves and the records in the daily returns are discussed. In both cases we try to describe the record statistics of the stock data with simple theoretical models. The daily returns are compared to i.i.d. RV's and the stock prices are modeled using a biased random walk, for which the record statistics are known. These models agree partly with the behavior of the stock data, but we also identify several interesting deviations. Most importantly, the number of records in the stocks appears to be systematically decreased in comparison with the random walk model. Considering the autoregressive AR(1) process, we can predict the record statistics of the daily stock prices more accurately. We also compare the stock data with simulations of the record statistics of the more complicated GARCH(1,1) model, which, in combination with the AR(1) model, gives the best agreement with the observational data. To better understand our findings, we discuss the survival and first-passage times of stock prices on certain intervals and analyze the correlations between the individual record events. After recapitulating some recent results for the record statistics of ensembles of N stocks, we also present some new observations for the weekly distributions of record events.Comment: 20 pages, 28 figure

Persistence of heavy-tailed sample averages occurs by infinitely many jumps

We consider the sample average of a centered random walk in $\mathbb{R}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set $A$ not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.Comment: 25 pages, 2 figure

Efficient Simulation and Conditional Functional Limit Theorems for Ruinous Heavy-tailed Random Walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed stochastic processes. Our changes-of-measure are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity -- a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain, as a corollary, functional conditional central limit theorems that extend classical results in the literature. We illustrate our methodology in the context of the ruin probability $P(\sup_n S_n >b)$, where $S_n$ is a random walk with heavy-tailed increments that have negative drift. Our techniques are based on the use of Lyapunov inequalities for variance control and termination time. The conditional limit theorems combine the application of Lyapunov bounds with coupling arguments

Correlated continuous-time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics

Standard continuous time random walk (CTRW) models are renewal processes in the sense that at each jump a new, independent pair of jump length and waiting time are chosen. Globally, anomalous diffusion emerges through action of the generalized central limit theorem leading to scale-free forms of the jump length or waiting time distributions. Here we present a modified version of recently proposed correlated CTRW processes, where we incorporate a power-law correlated noise on the level of both jump length and waiting time dynamics. We obtain a very general stochastic model, that encompasses key features of several paradigmatic models of anomalous diffusion: discontinuous, scale-free displacements as in Levy flights, scale-free waiting times as in subdiffusive CTRWs, and the long-range temporal correlations of fractional Brownian motion (FBM). We derive the exact solutions for the single-time probability density functions and extract the scaling behaviours. Interestingly, we find that different combinations of the model parameters lead to indistinguishable shapes of the emerging probability density functions and identical scaling laws. Our model will be useful to describe recent experimental single particle tracking data, that feature a combination of CTRW and FBM properties.Comment: 25 pages, IOP style, 5 figure