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This chapter derives asymptotic properties of the least squares (LS) estimator of the autoregressive (AR) parameter in local to unity processes with errors being fractional Gaussian noise (FGN) with the Hurst parameter H∈(0,1). It is shown that the estimator is consistent for all values of H∈(0,1). Moreover, the rate of convergence is n−1 when H∈[0.5,1). The rate of convergence is n−2H when H∈(0,0.5). Furthermore, the limiting distribution of the centered LS estimator depends on H. When H=0.5, the limiting distribution is the same as that obtained in Phillips (1987a) for the local to unity model with errors for which the standard functional central limit theorem is applicable. When H > 0.5 or when H < 0.5, the limiting distributions are new to the literature. The asymptotic properties of the LS estimator with fitted intercept are also derived. Simulation studies are performed to check the reliability of the asymptotic approximation for different values of sample size.

The Hurst exponent has been very important in telling the difference between fractal signals and explaining their significance. For estimators of the Hurst exponent, accuracy and efficiency are two inevitable considerations. The main purpose of this study is to raise the execution efficiency of the existing estimators, especially the fast maximum likelihood estimator (MLE), which has optimal accuracy.

A two-stage procedure combining a quicker method and a more accurate one to estimate the Hurst exponent from a large to small range will be developed. For the best possible accuracy, the data-induction method is currently ideal for the first-stage estimator and the fast MLE is the best candidate for the second-stage estimator.

For signals modeled as discrete-time fractional Gaussian noise, the proposed two-stage estimator can save up to 41.18 per cent the computational time of the fast MLE while remaining almost as accurate as the fast MLE, and even for signals modeled as discrete-time fractional Brownian motion, it can also save about 35.29 per cent except for smaller data sizes.

The proposed two-stage estimation procedure is a novel idea. It can be expected that other fields of parameter estimation can apply the concept of the two-stage estimation procedure to raise computational performance while remaining almost as accurate as the more accurate of two estimators.

The purpose of this paper is to test the efficient market hypothesis for major Indian sectoral indices by means of long memory approach in both time domain and frequency domain. This paper also tests the accuracy of the detrended fluctuation analysis (DFA) approach and the local Whittle (LW) approach by means of Monte Carlo simulation experiments.

The author applies the DFA approach for the computation of the scaling exponent in the time domain. The robustness of the results is tested by the computation of the scaling exponent in the frequency domain by means of the LW estimator. The author applies moving sub-sample approach on DFA to study the evolution of market efficiency in Indian sectoral indices.

The Monte Carlo simulation experiments indicate that the DFA approach and the LW approach provides good estimates of the scaling exponent as the sample size increases. The author also finds that the efficiency characteristics of Indian sectoral indices and their stages of development are dynamic in nature.

This paper has both methodological and empirical originality. On the methodological side, the author tests the small sample properties of the DFA and the LW approaches by using simulated series of fractional Gaussian noise and find that both the approach possesses superior properties in terms of capturing the scaling behavior of asset prices. On the empirical side, the author studies the evolution of long-range dependence characteristics in Indian sectoral indices.

The purpose of this paper is to introduce the performance of traditional collision resolution algorithm (CRA) under self‐similar traffic and present a prediction‐based CRA for wireless media access. With experimentations, the method is there after evaluated.

The traditional traffic models are mostly based on “Poisson” model or “Bernoulli” process, but recent decade traffic measurements found the coexistence of both long‐ and short‐range dependence in network traffic. On the other hand, CRA is an effective strategy to improve the performance of multiple access protocol and it achieves the highest capacity among all known multiple access protocols under the Poisson traffic model. In this paper, a CRA model is built on OPNET to study the effects of different traffic traces such as the fractional autoregressive integrated moving average process with non‐Gaussian white driving sequence and the real traffic data that are captured at a well‐attended ACM Conference. The performance is compared of traditional CRA based on the self‐similar traffic model and Poisson model and a novel CRA based on time – series prediction theory under self‐similar traffic models is designed.

The traditional Poisson traffic model gets the best performance under traditional CRA, while the self‐similar traffic performance under traditional CRA is too poor to be applied in an actual network environment. For example, the Poisson traffic model obtains the biggest throughput, the smallest delay and smallest collision resolution numbers under traditional CRA. This paper demonstrates the first‐come first‐serve (FCFS) must be improved under self‐similar traffic models. The novel collision resolution strategy‐based prediction can provide better performance under self‐similar traffic. The parameters of performance, such as throughput, delay and collision resolution numbers, are better than the traditional CRA.

This paper presents a prediction‐based CRA for self‐similar traffic which is the combination of the FCFS and the prediction theory and also provides a new method to resolve packets colliding.

In order to assess the informational efficiency of African equity markets (AEMs), the purpose of this paper is to examine long memory in both equity returns and volatility using auto‐regressive fractionally integrated moving average (ARFIMA)‐FIGARCH/hyperbolic GARCH (HYGARCH) models.

In order to test for long memory, the behaviour of the auto‐correlation function for 11 AEMs is examined. Following the graphical analysis, the authors proceed to estimate ARFIMA‐FIGARCH and ARFIMA‐HYGARCH models, specifically designed to capture long‐memory dynamics.

The results show that these markets (largely) display a predictable component in returns; while evidence of long memory in volatility is very mixed. In comparison, results from the control of the UK and USA show short memory in returns while evidence of long memory in volatility is mixed. These results show that the behaviour of equity market returns and risks are dissimilar across markets and this may have implications for portfolio diversification and risk management strategies.

The results of the analysis may have important implications for portfolio diversification and risk management strategies.

The importance of this paper lies in it being the first to systematically analyse long‐memory dynamics for a range of AEMs. African markets are becoming increasingly important as a source of international portfolio diversification and risk management. Hence, the results here have implication for the conduct of international portfolio building, asset pricing and hedging.

To define the main elements of a formal calculus which deals with fractional Brownian motion (fBm), and to examine its prospects of applications in systems science.

The approach is based on a generalization of the Maruyama's notation. The key is the new Taylor's series of fractional order f(x+h)=E_α(h^αD^α)f(x), where E_α( · ) is the Mittag‐Leffler function.

As illustrative applications of this formal calculus in systems science, one considers the linear quadratic Gaussian problem with fractal noises, the analysis of the equilibrium position of a system disturbed by a local fractal time, and a model of growing which involves fractal noises. And then, one examines what happens when one applies the maximum entropy principle to systems involving fBms (or shortly fractals).

The framework of this paper is applied mathematics and engineering mathematics, and the results so obtained allow the practical analysis of stochastic dynamics subject to fractional noises.

The direct prospect of application of this approach is the analysis of some stock markets dynamics and some biological systems.

The fractional Taylor's series is new and thus so are all its implications.

The purpose of this study is to originally present noise analysis of electrical circuits defined on fractal set.

The fractal integrodifferential equations of resistor-inductor, resistor-capacitor, inductor-capacitor and resistor-inductor-capacitor circuits subjected to zero mean additive white Gaussian noise defined on fractal set have been formulated. The fractal time component has also been considered. The closed form expressions for crucial stochastic parameters of circuit responses have been derived from these equations. Numerical simulations of power spectral densities based on the derived autocorrelation functions have been performed. A comparison with those without fractal time component has been made.

We have found that the Hausdorff dimension of the middle b Cantor set strongly affects the power spectral densities; thus, the average powers of noise induced circuit responses and the inclusion of fractal time component causes significantly different analysis results besides the physical measurability of electrical quantities.

For the first time, the noise analysis of electrical circuit on fractal set has been performed. This is also the very first time that the fractal time component has been included in the fractal calculus-based circuit analysis.

The paper aims to model multiple-period market risk forecasts under long memory persistence in market volatility.

The paper proposes volatility forecasts based on a combination of the GARCH(1,1)-model with potentially fat-tailed and skewed innovations and a long memory specification of the slowly declining influence of past volatility shocks. As the square-root-of-time rule is known to be mis-specified, the GARCH setting of Drost and Nijman is used as benchmark model. The empirical study of equity market risk is based on daily returns during the period January 1975 to December 2010. The out-of-sample accuracy of VaR predictions is studied for 5, 10, 20 and 60 trading days.

The long memory scaling approach remarkably improves VaR forecasts for the longer horizons. This result is only in part due to higher predicted risk levels. Ex post calibration to equal unconditional VaR levels illustrates that the approach also enhances efficiency in allocating VaR capital through time.

The improved VaR forecasts show that one should account for long memory when calibrating risk models.

The paper models single-period returns rather than choosing the simpler approach of modeling lower-frequency multiple-period returns for long-run volatility forecasting. The approach considers long memory in volatility and has two main advantages: it yields a consistent set of volatility predictions for various horizons and VaR forecasting accuracy is improved.

The theory which is addressed in this paper is that we should not be surprised to come across fractals in the analysis of some dynamic systems involving human factors. Moreover, in substance, fractals in human behaviour is acceptable. For the convenience of the reader a primary background to some models of fractional Brownian motions which can be found in the literature is given, and then the main features of the complex‐valued model, via a random walk in the complex plane, recently introduced by the author are recalled. The practical meaning of the model is exhibited. The parallel of the central limit theorem here is Levy’s stability. If it is supposed that human decision‐makers work via an observation process which combines the Heisenberg principle and a quantization principle in the measurement, then fractal dynamics appears to be quite in order. The relation with the theory of relative information is exhibited. The conjecture is then the following: could this model explain why fractals appear in finance, for instance?