Estudos empíricos [1],[2] demonstraram que o trafego das redes Internet Protocol (IP) possui propriedades fractais tais como impulsividade, auto-similaridade e dependência de longa duração em diversas escalas de agregação temporal, na faixa de milissegundos a minutos. Essas características tem motivado o desenvolvimento de novos modelos fractais de teletráfego e de novos algoritmos de controle de trafego em redes convergentes. Este trabalho propõe um novo modelo de trafego no espaço de estados baseado numa aproximação finito-dimensional do processo AutoRegressive Fractionally Integrated Moving Average (ARFIMA). A modelagem por meio de processos auto-regressivos (AR) também é investigada. A analise estatística de series simuladas e de series reais de trafego mostra que a aplicação de modelos AR de ordem alta em esquemas de previsão de teletráfego é fortemente prejudicada pelo problema da identificação da ordem do modelo. Também demonstra-se que a modelagem da memória longa pode ser obtida as custas do posicionamento de um ou mais pólos nas proximidades do circulo de raio unitário. Portanto, a implementação do modelo AR ajustado pode ser instável devido a efeitos de quantização dos coeficientes do filtro digital. O modelo de memória longa proposto oferece as seguintes vantagens: a) possibilidade de implementação pratica, pois não requer memória infinita, b) modelagem (explícita) da região das baixas freqüências do espectro e c) viabilização da utilização do filtro de Kalman. O estudo de caso apresentado demonstra que é possível aplicar o modelo de memória longa proposto em trechos estacionários de sinais de teletráfego fractal. Os resultados obtidos mostram que a dinâmica do parâmetro de Hurst de sinais de teletráfego pode ser bastante lenta na pratica. Sendo assim, o novo modelo proposto é adequado para esquemas de previsão de trafego, tais como Controle de Admissão de Conexões (CAC) e alocação dinâmica de banda, dado que o parâmetro de Hurst pode ser estimado em tempo real por meio da aplicação da transformada wavelet discreta (Discrete Wavelet Transform (DWT)). / Empirical studies [1],[2] demonstrated that heterogeneous IP traffic has fractal properties such as impulsiveness, self-similarity, and long-range dependence over several time scales, from miliseconds to minutes. These features have motivated the development of new traffic models and traffic control algorithms. This work presents a new state-space model for teletraffic which is based on a finite-dimensional representation of the ARFIMA random process. The modeling via AutoRegressive (AR) processes is also investigated. The statistical analysis of simulated time series and real traffic traces show that the application of high-order AR models in schemes of teletraffic prediction can be highly impaired by the model identification problem. It is also demonstrated that the modeling of the long memory can be obtained at the cost of positioning one or more poles near the unit circle. Therefore, the implementation of the adjusted AR model can be unstable due to the quantization of the digital filter coefficients. The proposed long memory model has the following advantages: a) possibility of practical implementation, inasmuch it does not require infinite memory, b) explicit modeling of the low frequency region of the power spectrum, and c) forecasts can be performed via the Kalman predictor. The presented case study suggests one can apply the proposed model in periods where stationarity can be safely assumed. The results indicate that the dynamics of the Hurst parameter can be very slow in practice. Hence, the new proposed model is suitable for teletraffic prediction schemes, such as CAC and dynamic bandwidth allocation, given that the Hurst parameter can be estimated on-line via DWT.

Long-range dependence in discrete and continuous time Markov chains over a countable state space is defined via embedded renewal processes brought about by visits to a fixed state. In the discrete time chain, solidarity properties are obtained and long-range dependence of functionals are examined. On the other hand, the study of LRD of continuous time chains is defined via the number of visits in a given time interval. Long-range dependence of Markov chains over a non-countable state space is also carried out through positive Harris chains. Embedded renewal processes in these chains exist via visits to sets of states called proper atoms. Examples of these chains are presented, with particular attention given to long-range dependent Markov chains in single-server queues, namely, the waiting times of GI/G/1 queues and queue lengths at departure epochs in M/G/1 queues. The presence of long-range dependence in these processes is dependent on the moment index of the lifetime distribution of the service times. The Hurst indexes are obtained under certain conditions on the distribution function of the service times and the structure of the correlations. These processes of waiting times and queue sizes are also examined in a range of M/P/2 queues via simulation (here, P denotes a Pareto distribution).

Today, the Internet is growing exponentially, with traffic statistics that mathematically exhibit fractal characteristics: self-similarity and long-range dependence. With these properties, data traffic shows high peak-to-average bandwidth ratios and causes networks inefficient. These problems make it difficult to predict, quantify, and control data traffic. In this thesis, two analytical methods are used to study fractal network traffic. They are second-order self-similarity analysis and multifractal analysis. First, self-similarity is an adaptability of traffic in networks. Many factors are involved in creating this characteristic. A new view of this self-similar traffic structure related to multi-layer network protocols is provided. This view is an improvement over the theory used in most current literature. Second, the scaling region for traffic self-similarity is divided into two timescale regimes: short-range dependence (SRD) and long-range dependence (LRD). Experimental results show that the network transmission delay separates the two scaling regions. This gives us a physical source of the periodicity in the observed traffic. Also, bandwidth, TCP window size, and packet size have impacts on SRD. The statistical heavy-tailedness (Pareto shape parameter) affects the structure of LRD. In addition, a formula to estimate traffic burstiness is derived from the self-similarity property. Furthermore, studies with multifractal analysis have shown the following results. At large timescales, increasing bandwidth does not improve throughput. The two factors affecting traffic throughput are network delay and TCP window size. On the other hand, more simultaneous connections smooth traffic, which could result in an improvement of network efficiency. At small timescales, in order to improve network efficiency, we need to control bandwidth, TCP window size, and network delay to reduce traffic burstiness. In general, network traffic processes have a Hlder exponent a ranging between 0.7 and 1.3. Their statistics differ from Poisson processes. From traffic analysis, a notion of the efficient bandwidth, EB, is derived. Above that bandwidth, traffic appears bursty and cannot be reduced by multiplexing. But, below it, traffic is congested. An important finding is that the relationship between the bandwidth and the transfer delay is nonlinear.

An estimation methodology to deal with fractionally integrated processes of Ornstein- Uhlenbeck type is proposed. The methodology is based on the continuous Whittle contrast. A simulation study is performed by driving this process with a symmetric CGMY background Lévy process.

Estudos empíricos [1],[2] demonstraram que o trafego das redes Internet Protocol (IP) possui propriedades fractais tais como impulsividade, auto-similaridade e dependência de longa duração em diversas escalas de agregação temporal, na faixa de milissegundos a minutos. Essas características tem motivado o desenvolvimento de novos modelos fractais de teletráfego e de novos algoritmos de controle de trafego em redes convergentes. Este trabalho propõe um novo modelo de trafego no espaço de estados baseado numa aproximação finito-dimensional do processo AutoRegressive Fractionally Integrated Moving Average (ARFIMA). A modelagem por meio de processos auto-regressivos (AR) também é investigada. A analise estatística de series simuladas e de series reais de trafego mostra que a aplicação de modelos AR de ordem alta em esquemas de previsão de teletráfego é fortemente prejudicada pelo problema da identificação da ordem do modelo. Também demonstra-se que a modelagem da memória longa pode ser obtida as custas do posicionamento de um ou mais pólos nas proximidades do circulo de raio unitário. Portanto, a implementação do modelo AR ajustado pode ser instável devido a efeitos de quantização dos coeficientes do filtro digital. O modelo de memória longa proposto oferece as seguintes vantagens: a) possibilidade de implementação pratica, pois não requer memória infinita, b) modelagem (explícita) da região das baixas freqüências do espectro e c) viabilização da utilização do filtro de Kalman. O estudo de caso apresentado demonstra que é possível aplicar o modelo de memória longa proposto em trechos estacionários de sinais de teletráfego fractal. Os resultados obtidos mostram que a dinâmica do parâmetro de Hurst de sinais de teletráfego pode ser bastante lenta na pratica. Sendo assim, o novo modelo proposto é adequado para esquemas de previsão de trafego, tais como Controle de Admissão de Conexões (CAC) e alocação dinâmica de banda, dado que o parâmetro de Hurst pode ser estimado em tempo real por meio da aplicação da transformada wavelet discreta (Discrete Wavelet Transform (DWT)). / Empirical studies [1],[2] demonstrated that heterogeneous IP traffic has fractal properties such as impulsiveness, self-similarity, and long-range dependence over several time scales, from miliseconds to minutes. These features have motivated the development of new traffic models and traffic control algorithms. This work presents a new state-space model for teletraffic which is based on a finite-dimensional representation of the ARFIMA random process. The modeling via AutoRegressive (AR) processes is also investigated. The statistical analysis of simulated time series and real traffic traces show that the application of high-order AR models in schemes of teletraffic prediction can be highly impaired by the model identification problem. It is also demonstrated that the modeling of the long memory can be obtained at the cost of positioning one or more poles near the unit circle. Therefore, the implementation of the adjusted AR model can be unstable due to the quantization of the digital filter coefficients. The proposed long memory model has the following advantages: a) possibility of practical implementation, inasmuch it does not require infinite memory, b) explicit modeling of the low frequency region of the power spectrum, and c) forecasts can be performed via the Kalman predictor. The presented case study suggests one can apply the proposed model in periods where stationarity can be safely assumed. The results indicate that the dynamics of the Hurst parameter can be very slow in practice. Hence, the new proposed model is suitable for teletraffic prediction schemes, such as CAC and dynamic bandwidth allocation, given that the Hurst parameter can be estimated on-line via DWT.

A set of observations from a random process which exhibit correlations that decay slower than an exponential rate is regarded as long-range dependent. This phenomenon has stimulated great interest in the scientific community as it appears in a wide range of areas of knowledge. For example, this property has been observed in data pertaining to electronics, econometrics, hydrology and biomedical signals.There exist several estimation methods for finding model parameters that help explain the set of observations exhibiting long-range dependence. Among these methods, maximum likelihood is attractive, given its desirable statistical properties such as asymptotic consistency and efficiency. However, its computational complexity makes the implementation of maximum likelihood prohibitive.This thesis presents a group of computationally efficient estimators based on the maximum likelihood framework. The thesis consists of two main parts. The first part is devoted to developing a computationally efficient alternative to the maximum likelihood estimate. This alternative is based on the circulant embedding concept and it is shown to maintain the desirable statistical properties of maximum likelihood.Interesting results are obtained by analysing the circulant embedding estimate. In particular, this thesis shows that the maximum likelihood based methods are ill-conditioned; the estimators' performance will deteriorate significantly when the set of observations is corrupted by errors. The second part of this thesis focuses on developing computationally efficient estimators with improved performance under the presence of errors in the observations.

Traditional financial modeling is based on semimartingale processes with stationary and independent increments. However, empirical investigations on financial data does not always support these assumptions. This contradiction showed that there is a need for new stochastic models. Fractional Brownian motion (fBm) was proposed as one of these models by Benoit Mandelbrot. FBm is the only continuous Gaussian process with dependent increments. Correlation between increments of a fBm changes according to its self-similarity parameter H. This property of fBm helps to capture the correlation dynamics of the data and consequently obtain better forecast results. But for values of H different than 1/2, fBm is not a semimartingale and classical Ito formula does not exist in that case. This gives rise to need for using the white noise theory to construct integrals with respect to fBm and obtain fractional Ito formulas. In this thesis, the representation of fBm and its fundamental properties are examined. Construction of Wick-Ito-Skorohod (WIS) and fractional WIS integrals are investigated. An Ito type formula and Girsanov type theorems are stated. The financial applications of fBm are mentioned and the Black&amp / Scholes price of a European call option on an asset which is assumed to follow a geometric fBm is derived. The statistical aspects of fBm are investigated. Estimators for the self-similarity parameter H and simulation methods of fBm are summarized. Using the R/S methodology of Hurst, the estimations of the parameter H are obtained and these values are used to evaluate the fractional Black&amp / Scholes prices of a European call option with different maturities. Afterwards, these values are compared to Black&amp / Scholes price of the same option to demonstrate the effect of long-range dependence on the option prices. Also, estimations of H at different time scales are obtained to investigate the multiscaling in financial data. An outlook of the future work is given.

For a stationary two dimensional random field, we utilize the classical Kolmogorov-Wiener theory to develop prediction methodology which requires minimal assumptions on the dependence structure of the random field. We also provide solutions for several non-standard prediction problems which deals with the "modified past," in which a finite number of observations are added to the past. These non-standard prediction problems are motivated by the network site selection in the environmental and geostatistical applications. Unlike the time series situation, the prediction results for random fields seem to be expressible only in terms of the moving average parameters, and attempts to express them in terms of the autoregressive parameters lead to a new and mysterious projection operator which captures the nature of edge-effects. We put forward an approach for estimating the predictor coefficients by carrying out an extension of the exponential models. Through simulation studies and real data example, we demonstrate the impressive performance of our prediction method. To the best of our knowledge, the proposed method is the first to deliver a unified framework for forecasting random fields both in the time and spectral domain without making a subjective choice of the covariance structure. Finally, we focus on the estimation of the hurst parameter for long range dependence stationary random fields, which draws its motivation from applications in the environmental and atmospheric processes. Current methods for estimation of the Hurst parameter include parametric models like fractional autoregressive integrated moving average models, and semiparametric estimators which are either inefficient or inconsistent. We propose a novel semiparametric estimator based on the fractional exponential spectrum. We develop three data-driven methods which can automatically select the optimal model order for the fractional exponential models. Extensive simulation studies and analysis of Mercer and Hall?s wheat data are used to illustrate the performance of the proposed estimator and model order selection criteria. The results show that our estimator outperforms existing estimators, including the GPH (Geweke and Porter-Hudak) estimator. We show that the proposed estimator is consistent, works for different definitions of long range dependent random fields, is computationally simple and is not susceptible to model misspecification or poor efficiency.

The thesis is made up of a number of studies involving long-range dependence (LRD), that is, a slow power-law decay in the temporal correlation of stochastic models. Such a phenomenon has been frequently observed in practice. The models with LRD often yield non-standard probabilistic and statistical results. The thesis includes in particular the following topics: Multivariate limit theorems. We consider a vector made of stationary sequences, some components of which have LRD, while the others do not. We show that the joint scaling limits of the vector exhibit an asymptotic independence property. Non-central limit theorems. We introduce new classes of stationary models with LRD through Volterra-type nonlinear filters of white noise. The scaling limits of the sum lead to a rich class of non-Gaussian stochastic processes defined by multiple stochastic integrals. Limit theorems for quadratic forms. We consider continuous-time quadratic forms involving continuous-time linear processes with LRD. We show that the scaling limit of such quadratic forms depends on both the strength of LRD and the decaying rate of the quadratic coefficient. Behavior of the generalized Rosenblatt process. The generalized Rosenblatt process arises from scaling limits under LRD. We study the behavior of this process as its two critical parameters approach the boundaries of the defining region. Inference using self-normalization and resampling. We introduce a procedure called "self-normalized block sampling" for the inference of the mean of stationary time series. It provides a unified approach to time series with or without LRD, as well as with or without heavy tails. The asymptotic validity of the procedure is established.

Using the ARFIMA-FIGARCH model, this dissertation examines the efficiency of Botswana's bond market. It focuses on the properties of the return and volatility of the Fleming Asset Bond Index (the main aggregate fixed income benchmark index in Botswana) over the period September 2009 to May 2019. The weak-form version of efficient market hypothesis (EMH) is used as a criterion to investigate the existence of long memory in both bond returns and volatility. The results of our study indicate that the Botswana bond market data follow, to a great extent, the long-range dependence which negates the precepts of the efficient market hypothesis. Furthermore, policy reforms intended to stimulate bond market reform and related efficiency gains appear not to have produced the desired outcomes as the existence of long memory is found across all sample periods. Further remedial policies are suggested to enhance market dynamism.