Marek Teuerle

Multidimensional Levy walk and its scaling limits

In this paper we obtain the scaling limit of a multidimensional Levy walk and describe the detailed structure of the limiting process. The scaling limit is a subordinated α-stable Levy motion with the parent process and subordinator being strongly dependent processes. The corresponding Langevin picture is derived. We also introduce a useful method of simulating Levy walks with a predefined spectral measure, which controls the direction of each jump. Our approach can be applied in the analysis of real-life data—we are able to recover the spectral measure from the data and obtain the full characterization of a Levy walk. We also give examples of some useful spectral measures, which cover a large class of possible scenarios in the modeling of real-life phenomena.

Modeling anomalous diffusion by a subordinated fractional Lévy-stable process

Two phenomena that can be discovered in systems with anomalous diffusion are long-range dependence and trapping events. The first effect concerns events that are arbitrarily distant but still influence each other exceptionally strongly, which is characteristic for anomalous regimes. The second corresponds to the presence of constant values of the underlying process. Motivated by the relatively poor class of models that can cover these two phenomena, we introduce subordinated fractional L?vy-stable motion with tempered stable waiting times. We present in detail its main properties, propose a simulation scheme and give an estimation procedure for its parameters. The last part of the paper is a presentation, via the Monte Carlo approach, of the effectiveness of the estimation of the parameters.

Dynamics of carbon dioxide concentration in indoor air

Abstract Carbon dioxide is an indicator of indoor air quality. A number of factors influence its concentration. Due to the fact that they all present time variability, CO2 concentration indoors considerably varies over time. In this work we focus on the dynamics of indoor CO2 concentration changes. We examine the dynamics of CO2 variation and use it as a source of information on the character of collective impact of factors on indoor air. The proposed method is based on mean square displacement analysis (MSD) applied to the segments of the time series of CO2 monitoring data. The segments are determined based on the introduced criterion for optimal sample size selection. The method was validated by showing that it reproduces the known stochastic dynamics of the simulated data set properly. From the real data analysis, we found that indoors the stochastic dynamics of CO2 concentration in time was mainly nonlinear. Moreover, it exhibited a cycle of change which could be associated with the daily variation of the collective influence of factors on indoor air. We intend to apply the method to other parameters of indoor air, aiming at developing a capability of describing the dynamics of indoor air as a complex system.

Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement

The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.

Ruin probability in finite time

The ruin probability in finite time can only be calculated analytically for a few special cases of the claim amount distribution. The most classic example is discussed in Section 1.2. The value can always be computed directly using Monte Carlo simulations, however, this is usually a time-consuming procedure. Thus, finding a reliable approximation is really important from a practical point of view. The most important approximations of the finite time ruin probability are presented in Section 1.3. They are further illustrated in Section 1.4 using the Danish fire losses dataset, which concerns major fire losses in profits that occurred between 1980 and 2002 and were recorded by Copenhagen Re.

Comparison of the two-power-law generalized Mittag-Leffler and Havriliak-Negami dielectric relaxation responses

In this paper analysis of the two-power-law relaxation behavior in terms of the generalized Mittag-Leffler (GML) and Havriliak-Negami (HN) responses is presented. Using the probabilistic representation of the relaxation function, differences and similarities between the GML and HN patterns are investigated by means of the Monte Carlo technique.

Measures of Dependence for -Stable Distributed Processes and Its Application to Diagnostics of Local Damage in Presence of Impulsive Noise

Local damage detection in rotating machinery is simply searching for cyclic impulsive signal in noisy observation. Such raw signal is mixture of various components with specific properties (deterministic, random, cyclic, impulsive, etc.). The problem appears when the investigated process is based on one of the heavy-tailed distributions. In this case the classical measure can not be considered. Therefore, alternative measures of dependence adequate for such processes should be considered. In this paper we examine the structure of dependence of alpha-stable based systems expressed by means of two measures, namely, codifference and covariation. The reason for using alpha-stable distribution is simple and intuitive: signal of interest is impulsive so its distribution is heavy-tailed. The main goal is to introduce a new technique for estimation of covariation. Due to the complex nature of such vibration signals applying novel methods instead of classical ones is recommended. Classical algorithms usually are based on the assumption that theoretical second moment is finite, which is not true in case of the data acquired on the faulty components. Main advantage of our proposed algorithm is independence from second moment assumption.