A major issue in time series analysis and particularly in the study of meteorological time series behaviour is the long range dependence (LRD). Various estimators of LRD have been proposed. Their accuracy have been generally tested by using simulated time series since sometimes only their asymptotic property are known, or worse, no asymptotic property have been proved. It is well – known that t...

We investigate the local fractal properties of the financial time series based on the evolution of the Warsaw Stock Exchange Index (WIG) connected with the largest developing financial market in Europe. Calculating the local Hurst exponent for the WIG time series we find an interesting dependence between the behavior of the local fractal properties of the WIG time series and the crashes appeara...

This paper presents steps towards creating a general model for aggregated traffic streams. We show that fitting the mean, variance and Hurst parameter is insufficient to consistently model a long range dependent traffic stream. A fourth parameter, the “level of aggregation,” is required. We also show that with increased aggregation, the behaviour of a traffic stream tends towards that of a frac...

We study the small deviation problem for a class of symmetric Lévy processes, namely, subordinated Lévy processes. These processes can be represented as W ◦A, where W is a standard Brownian motion, and A is a subordinator independent of W . Under some mild general assumption, we give precise estimates (up to a constant multiple in the logarithmic scale) of the small deviation probabilities. The...

We give a stochastic microscopic modelling of stock markets driven by continuous double auction. If we take into account the mimetic behavior of traders, when they place limit order, our virtual markets shows the power-law tail of the distribution of returns with the exponent outside the Levy stable region, the short memory of returns and the long memory of volatilities. The Hurst exponent of o...

We study the possible link between “local turbulence strength” in a flow which is represented by a finite time series and a “chaotic invariant”, namely, the leading Lyaponuv exponent that characterizes this series. To validate a conjecture about this link, we analyze several time series of measurements taken by a plane flying at constant height in the upper troposphere. For each of these time s...

This paper describes some of the results in [5] for a stochastic calculus for a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1). Two stochastic integrals are defined with explicit expressions for their first two moments. Multiple and iterated integrals of a fractional Browinian motion are defined and various properties of these integrals are given. A square integrab...

The article is devoted to analysis of currency quotes behavior on the currency market by defining dynamic changes over time. The main tool of fractal analysis is the Hurst under the hypothesis of fractal market. For 17 major currency pairs on closing prices and the prices maximum-minimum the Hurst index is calculated by formula for the adjusted R/S analysis. Values at the currency markets of di...

We investigate the departure process of an ATM output buffer intaking asymptotically second-order self-similar traffic using matrix-analytic technique. The results obtained include the marginal and joint distribution of interdeparture times. We show that output and input processes asymptotically possess the same tail and smoothing effect takes place only for traffic with small Hurst parameters.

We consider the class of pentagonal 3-polytopal graphs all of whose edges are incident either with two 3-valent vertices or with a 3-valent vertex and a q-valent vertex. For most values of q, (i) we find a small non-hamiltonian graph in the class and (ii) we show that the shortness exponent of the class and the shortness coefficient of a special subclass are less than one. For q--4, we find a p...