Domain of attraction

This note is an answer to some open problems connected with recent developments for appropriate methodologies for making inferences on the tail of a distribution function (d.f.). Namely, in Fraga Alves and Gomes (1996), the Gumbel statistic, based on the top part of a sample, is used in a semi-parametric approach, in order to fit an appropriate tail to the underlying model to a data set. The problem of statistical inference about extremal observations is handled there according to a test for choosing the most appropriate domain of attraction for the tail distribution, which gives preference to the Gumbel domain for the null hypothesis. The asymptotic behaviour of the referred statistic is derived therein under that null hypothesis and here we present similar extended results under the alternative conditions, i.e., for d.f. that belongs to the other Generalized Extreme Value domains, as an accomplishment to the promise made in last chapters of Fraga Alves and Gomes (1995; 1996).

Let (X jk ) j,k 1 be i.i.d. complex random variables such that X jk is in the domain of attraction of an α-stable law, with 0 < α < 2. Our main result is a heavy tailed counterpart of Girko's circular law. Namely, under some additional smoothness assumptions on the law of X jk , we prove that there exist a deterministic sequence an ∼ n 1/α and a probability measure µα on C depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix (a −1 n X jk ) 1 j,k n converges weakly to µα as n → ∞. Our approach combines Aldous & Steele's objective method with Girko's Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of µα. In contrast with the Hermitian case, we find that µα is not heavy tailed.

A, VANNELLII" and M. VIDYASAGAR~ A new concept known as a maximal Lyapunov function, based on rational Lyapunov functions rather than polynominals, can compute the domain of attraction exactly using a new iterative procedure for estimating the domain of attraction.

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt . In the particular case of a subordinator in the domain of attraction of a stable distribution our results are the analogue of the results obtained by the authors in [6] for non-monotone Lévy processes. For subordinators an approach different to that in [6] is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential. ‡ Research funded by the CONACYT Project Teoría y aplicaciones de procesos de Lévy where b denotes the drift and Π the Lévy measure of X. We will write ψ * for the exponent of {Xt − bt, t ≥ 0}, so that ψ * (λ) := ψ(λ) − bλ, λ ≥ 0.

In this paper we show that the continuous version of the self normalised process $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ where $S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}= \sum_{i=1}^{n}|X_i|^p)^{\frac{1}{p}}$ and $X_i$ i.i.d. random variables belong to $DA(\alpha)$, has a non trivial distribution iff $p=\alpha=2$. The case for $2 > p > \alpha$ and $p \le \alpha < 2$ is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Cs\"org\"o et al. who showed Donsker's theorem for $Y_{n,2}(\cdot)$, i.e., for $p=2$, holds iff $\alpha =2$ and identified the limiting process as standard Brownian motion in sup norm.

The emergence of CDS indices and corresponding credit risk transfer markets with high liquidity and narrow bid-ask spreads has created standard benchmarks for market credit risk and correlation against which portfolio credit risk models can be calibrated. Integrated risk management for correlation dependent credit derivatives, such as single-tranches of synthetic CDOs, requires an approach that adequately reflects the joint default behavior in the underlying credit portfolios. Another important feature for such applications is a flexible model architecture that incorporates the dynamic evolution of underlying credit spreads. In this paper, we present a model that can be calibrated to quotes of CDS index-tranches in a statistically sound way and simultaneously has a dynamic architecture to provide for the joint evolution of distance-to-default measures. This is accomplished by replacing the normal distribution by smoothly truncated α-stable (STS) distributions in the Black/Cox version of the Merton approach for portfolio credit risk. This is possible due to the favorable features of this distribution family, namely, consistent application in the Black/Scholes no-arbitrage framework and the preservation of linear correlation concepts. The calibration to spreads of CDS index tranches is accomplished by a genetic algorithm. Our distribution assumption reflects the observed leptokurtic and asymmetric properties of empirical asset returns since the STS distribution family is basically constructed from α-stable distributions. These exhibit desirable statistical properties such as domains of attraction and the application of the generalized central limit theorem. Moreover, STS distributions fulfill technical restrictions like finite (exponential) moments of arbitrary order. In comparison to the performance of the basic normal distribution model which lacks tail dependence effects, our empirical analysis suggests that our extension with a heavy-tailed and highly peaked distribution provides a better fit to tranche quotes for the iTraxx IG index. Since the underlying implicit modeling of the dynamic evolution of credit spreads leads to such results, this suggests that the proposed model is appropriate to price and hedge complex transactions that are based on correlation dependence. A further application might be integrated risk management activities in debt portfolios where concentration risk is dissolved by means of portfolio credit risk transfer instruments such as synthetic CDOs.

Image guidance promises to improve targeting accuracy and broaden the scope of medical procedures performed with needles. This paper takes a step toward automating the guidance of a flexible tipsteerable needle as it is inserted into human tissue. We build upon a previously proposed nonholonomic model of needles that derive steering from asymmetric bevel forces at the tip. The bevel-tip needle is inserted and rotated at its base in order to steer it in six degrees of freedom. As a first step for control, we show that the needle tip can be automatically guided to a planar slice of tissue as it is inserted. Our approach keeps the physician in the loop to control insertion speed. The distance of the needle tip position from the plane of interest is used to drive an observer-based feedback controller which we prove is locally asymptotically stable. Numerical simulations demonstrate a large domain of attraction and robustness of the controller in the face of parametric uncertainty and measurement noise. Physical experiments with tip-steerable Nitinol needles inserted into a transparent plastisol tissue phantom under stereo image guidance validate the effectiveness of our approach.

A convex approach to robust regional stability analysis of a class of nonlinear state-delayed systems subject to convex-bounded parameter uncertainty is proposed. Delay-dependent conditions are developed to ensure system robust local stability and obtain an estimate of a domain of attraction of the origin inside a given polytopic region of the state-space. This approach is then extended to provide a delay-dependent solution to the problem of L 2 -gain analysis. The proposed approach is based on a Lyapunov-Krasovskii functional with polynomial dependence on the system state and uncertain parameters and is formulated in terms of linear matrix inequalities. Numerical examples illustrate the potentials of the derived results.

This paper proposes a strategy for estimating the DA (domain of attraction) for non-polynomial systems via LFs (Lyapunov functions). The idea consists of converting the non-polynomial optimization arising for a chosen LF in a polynomial one, which can be solved via LMI optimizations. This is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst-case remainders in truncated Taylor expansions. Moreover, a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen LF, and another condition is provided for establishing whether such an estimate has been found. The proposed strategy can readily be exploited with variable LFs in order to search for optimal estimates. Lastly, it is worth to remark that no other method is available to estimate the DA for non-polynomial systems via LMIs.

The spring-loaded inverted pendulum (SLIP), or monopedal hopper, is an archetypal model for running in numerous animal species. Although locomotion is generally considered a complex task requiring sophisticated control strategies to account for coordination and stability, we show that stable gaits can be found in the SLIP with both linear and "air" springs, controlled by a simple fixed-leg reset policy. We first derive touchdown-to-touchdown Poincaré maps under the common assumption of negligible gravitational effects during the stance phase. We subsequently include and assess these effects and briefly consider coupling to pitching motions. We investigate the domains of attraction of symmetric periodic gaits and bifurcations from the branches of stable gaits in terms of nondimensional parameters.

The increased use of DC microgrid for critical application lead to the necessity of advanced control design for a stable operation of the system. The loads connected to DC microgrid are controlled with power electronic devices and shows the behaviour of constant power load (CPL), which poses a serious challenge to stability as it adds nonlinearity and minimises the effective damping. This paper presents a robust controller design approach based on Hardy Space (H)-infinity control norms to tackle the nonlinearity added by CPL by expanding the region for stability. The design criteria are based on Lyapunov theory of nonlinear systems within the framework of Linear Matrix Inequality (LMI). The necessary and sufficient conditions are obtained in terms of linear matrix inequalities to ensure the transient performance and stability of the system. The LMI equation is solved to maximize the size of the estimated domain of stability. The performance of the proposed controller is verified using simulation in MATLAB/Simulink. The DC microgrid in this paper consists of a solar photovoltaic (PV) unit and a battery as an energy storage system together with loads. The Proposed controller not only ensures the stability of the system but also guarantees the improved transient performance of the closed loop system by expanding the size of the stability region.

This paper discusses two optimal allocation problems. We consider different hypotheses of portfolio selection with stable distributed returns for each of them. In particular, we study the optimal allocation between a riskless return and risky stable distributed returns. Furthermore, we examine and compare the optimal allocation obtained with the Gaussian and the stable non-Gaussian distributional assumption for the risky return. KEY WORDS: optimal allocation, stochastic dominance, risk aversion, measure of risk, a stable distribution, domain of attraction, sub-Gaussian stable distributed, fund separation, normal distribution, mean variance analysis, safety-first analysis.

We use a recently developed duality theory for linear differential inclusions (LDIs) to enhance the stability analysis of systems with saturation. Based on the duality theory, the condition of stability for a LDI in terms of one Lyapunov function can be easily derived from that in terms of a Lyapunov function conjugate to the original one in the sense of convex analysis. This paper uses a particular conjugate pair, the convex hull of quadratics and the maximum of quadratics, along with their dual relationship, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a LDI system with an effective approach which enables optimization on the local LDI description. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates that the estimation of the domain of attraction by this paper's methods drastically improve those by the earlier methods.

This paper proposes several parametric models to compute the portfolio VaR and CVaR in a given temporal horizon and for a given level of confidence. Firstly, we describe extension of the EWMA RiskMetrics model considering conditional elliptically distributed returns. Secondly, we examine several new models based on different stable Paretian distributional hypotheses of return portfolios. Finally, we discuss the applicability

We consider the dynamics of a general stage-structured predator-prey model which generalizes several known predator-prey, SEIR, and virus dynamics models, assuming that the intrinsic growth rate of the prey, the predation rate, and the removal functions are given in an unspecified form. Using the Lyapunov method, we derive sufficient conditions for the local stability of the equilibria together with estimations of their respective domains of attraction, while observing that in several particular but important situations these conditions yield global stability results. The biological significance of these conditions is discussed and the existence of the positive steady state is also investigated.

This paper investigates PID control design for a class of planar nonlinear uncertain systems in the presence of actuator saturation. Based on the bounds on the growth rates of the nonlinear uncertain function in the system model, the system is placed in a linear differential inclusion. Each vertex system of the linear differential inclusion is a linear system subject to actuator saturation. By placing the saturated PID control into a convex hull formed by the PID controller and an auxiliary linear feedback law, we establish conditions under which an ellipsoid is contractively invariant and hence is an estimate of the domain of attraction of the equilibrium point of the closed-loop system. The equilibrium point corresponds to the desired set point for the system output. Thus, the location of the equilibrium point and the size of the domain of attraction determine, respectively, the set point that the output can achieve and the range of initial conditions from which this set point can be reached. Based on these conditions, the feasible set points can be determined and the design of the PID control law that stabilizes the nonlinear uncertain system at a feasible set point with a large domain of attraction can then be formulated and solved as a constrained optimization problem with constraints in the form of linear matrix inequalities (LMIs). Application of the proposed design to a magnetic suspension system illustrates the design process and the performance of the resulting PID control law.

Model predictive control (MPC) is one of the few techniques which is able to handle with constraints on both state and input of the plant. The admissible evolution and asymptotically convergence of the closed loop system is ensured by means of a suitable choice of the terminal cost and terminal contraint. However, most of the existing results on MPC are designed for a regulation problem. If the desired steady state changes, the MPC controller must be redesigned to guarantee the feasibility of the optimization problem, the admissible evolution as well as the asymptotic stability. In this paper a novel formulation of the MPC is proposed to track varying references. This controller ensures the feasibility of the optimization problem, constraint satisfaction and asymptotic evolution of the system to any admissible steady-state. Hence, the proposed MPC controller ensures the offset free tracking of any sequence of piece-wise constant admissible set points. Moreover this controller requires the solution of a single QP at each sample time, it is not a switching controller and improves the performance of the closed loop system. Copyright©2005 IFAC.

The literature disagrees about the statistical distribution of snow avalanche crown depths. Large datasets from Mammoth Mountain, California and the Westwide Avalanche Network show that the three-parameter generalized extreme value distribution provides the most robust fit, followed by a two-parameter variation, the Fréchet distribution. The most parsimonious explanation is neither self-organized criticality nor other complex cascades, but the maximum domain of attraction, implying that distributions of individual avalanche crown depths are scaling. We also show that crown depths do not have a universal tail index. Rather, they range from 2.8 to 4.6 over different avalanche paths, consistent with other geophysical phenomena such as wildfires, which show similar variability.

We propose an estimator of the Pareto tail index m of a distribution, that competes well with the Hill, Pickands and moment estimators. Unlike the above estimators, that are based only on the extreme observations, the proposed estimator uses all observations; its idea rests in the tail behavior of the sample mean ¯ Xn, having a simple structure under heavy-tailed F. The observations, partitioned into N independent samples of sizes n, lead to N sample means whose empirical distribution function is the main estimation tool. The estimator is strongly consistent and asymptotically normal as N →∞ , while n remains fixed. Its behavior is illustrated in a simulation study.

Estimating the Domain of Attraction (DA) of equilibrium points is a problem of fundamental importance in systems engineering. Several approaches have been proposed for the case of known polynomial systems allowing one to find the Largest Estimate of the DA (LEDA) for a given Lyapunov Function (LF). However, the problem of estimating the Robust DA (RDA), that is the DA guaranteed for all possible uncertainties in an uncertain system, it is still an unsolved problem. In this paper, LMI methods are proposed for estimating the RDA in the case of systems depending polynomially in the state and in the uncertainty which is supposed to belong to a polytope. Specifically, the issue of computing the Robust LEDA (RLEDA), that is the intersection of all LEDAs, is considered for common and parameter-dependent LFs, providing constant and parameter-dependent lower bounds. The computation of approximations with simple shape of the RLEDA in the case of parameter-dependent LFs is also discussed.

This paper considers the design of output tracking systems subject to actuator saturation and integrator windup. An optimization-based approach is developed to design feedback and anti-windup gains of a controller structure involving intelligent integrators. The design goal is to increase stability region and output tracking and disturbance rejection ability of the closed-loop system. ?

After more than 40 years the so-called Hurst effect remains an open problem in stochastic hydrology. Historically, its existence has been explained either by preasymptotic behavior of the rescaled adjusted range R* n , certain classes of nonstationarity in time series, infinite memory, or erroneous estimation of the Hurst exponent. Various statistical tests to determine whether an observed time series exhibits the Hurst effect are presented. The tests are based on the fact that for the family of processes in the Brownian domain of attraction, R* n /((0n)) 1/2 converges in distribution to a nondegenerate random variable with known distribution (functional central limit theorem). The scale of fluctuation 0, defined as the sum of the correlation function, plays a key role. Application of the tests to several geophysical time series seems to indicate that they do not exhibit the Hurst effect, although those series have been used as examples of its existence, and furthermore the traditional pox diagram method to estimate the Hurst exponent gives values larger than 0.5. It turned out that the coefficient in the relation of/?* versus n, which is directly proportional to the scale of fluctuation, was more important than the exponent. The Hurst effect motivated the popularization of I//noises and related ideas of fractals and scaling. This work illustrates how delicate the procedures to deal with infinity must be.

... Secondly, many problems of practical interest are non-monotone, due to interactions at uncontrolled intersections, responsive signals, or mixed-mode interactions (see Watling, 1996, for examples). In such cases, multiple equilibria may exist, some of which may be unstable (ie ...

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

We present a method to estimate the domain of attraction for a discrete-time linear system under a saturated linear feedback. A simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant. Moreover, the condition can be expressed as linear matrix inequalities (LMIs) in terms of all the varying parameters and hence can easily be used for controller synthesis. The following surprising result is revealed for systems with single input: suppose that an ellipsoid is made invariant with a linear feedback, then it is invariant under the saturated linear feedback if and only if there exists a saturated (nonlinear) feedback which makes the ellipsoid invariant. Finally, the set invariance condition is extended to determine invariant sets for systems with persistent disturbances. LMI based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements.

This paper discusses two optimal allocation problems. We consider different hypotheses of portfolio selection with stable distributed returns for each of them. In particular, we study the optimal allocation between a riskless return and risky stable distributed returns. Furthermore, we examine and compare the optimal allocation obtained with the Gaussian and the stable non-Gaussian distributional assumption for the risky return.

The problem of regulating the output voltage of the Boost DC-to-DC power converter has attracted the attention of many control researchers for several years now. Besides its practical relevance, the system is an interesting theoretical case study because it is a switched device whose averaged dynamics are described by a bilinear second order nonminimum phase system with saturated input, partial state measurement and a highly uncertain parameter-the load resistance. In this paper we propose an output-feedback saturated controller which ensures regulation of the desired output voltage and is, at the same time, insensitive to uncertainty in the load resistance. Furthermore, bounds on this parameter can be used to tune the controller so as to (locally) ensure robust performance, e.g., that the transient has no (under) overshoot. The controller, which is designed following the passivity-based interconnection and damping assignment methodology recently proposed in the literature, is a static nonlinear output feedback. This allows us to invoke simple phase-plane techniques to determine the exact domain of attraction as well as the admissible initial conditions that ensure the control objectives. One ÿnal advantage of our controller is that it is computationally less demanding than the industry standard lead-lag ÿlters.

Let X be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function c. As an analogue of the random walk results in and we study the local behaviour of the distribution of the lifetime ζ under the characteristic measure n of excursions away from 0 of the process X reflected in its past infimum, and of the first passage time of X below 0, T0 = inf{t > 0 : Xt < 0}, under Px(·), for x > 0, in two different regimes for x, viz. x = o(c(·)) and x > Dc(·), for some D > 0. We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at T0 and discontinuous passage. In the way to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to X reflected in its past infimum. ∞ 0 µ * (x)dx. (See [4]

This paper provides a method for designing a linear state feedback control law, with the following specifications. Given a polytope which surround the origin of the state space, the design procedure is aimed to define a controller such that the zero equilibrium point of the closed loop system is asymptotically stable and a chosen polytope belongs to the domain of attraction of the zero equilibrium. The control problem is solved in terms of some Linear Matrix Inequalities. An example shows the practical applicability of the proposed procedure in the control of a robot wrist.

We consider the random reversible Markov kernel K obtained by assigning i.i.d. nonnegative weights to the edges of the complete graph over n vertices and normalizing by the corresponding row sum. The weights are assumed to be in the domain of attraction of an αstable law, α ∈ (0, 2). When 1 ≤ α < 2, we show that for a suitable regularly varying sequence κn of index 1 − 1/α, the limiting spectral distribution µα of κnK coincides with the one of the random symmetric matrix of the un-normalized weights (Lévy matrix with i.i.d. entries). In contrast, when 0 < α < 1, we show that the empirical spectral distribution of K converges without rescaling to a nontrivial law µα supported on [−1, 1], whose moments are the return probabilities of the random walk on the Poisson weighted infinite tree (PWIT) introduced by Aldous. The limiting spectral distributions are given by the expected value of the random spectral measure at the root of suitable self-adjoint operators defined on the PWIT. This characterization is used together with recursive relations on the tree to derive some properties of µα and µα. We also study the limiting behavior of the invariant probability measure of K.

This paper proposes several parametric models to compute the portfolio VaR and CVaR in a given temporal horizon and for a given level of confidence. Firstly, we describe extension of the EWMA RiskMetrics model considering conditional elliptically distributed returns. Secondly, we examine several new models based on different stable Paretian distributional hypotheses of return portfolios. Finally, we discuss the applicability

In this paper, we present a new output feedback control approach for discrete-time linear systems subject to actuator saturations using parameter-dependent Lyapunov functions. The saturation level indicator serves as a scheduling parameter. The resulting nonlinear controller is expressed in a quasi-LPV (linear parameter-varying) form, and the stabilization and disturbance attenuation problems are formulated and solved as finite-dimensional linear matrix inequality (LMI) optimization problems. Our approach is less conservative than a single quadratic Lyapunov function method. Specifically, the proposed output feedback control law asymptotically stabilizes the open loop system with a larger domain of attraction and achieves better disturbance attenuation under energy and magnitude bounded disturbances.

The existence of travelling fronts and their uniqueness modulo translations are proved in the context of a one dimensional, non local, evolution equation derived in [5] from Ising systems with Glauber dynamics and Kac potentials. The front describes the moving interface between the stable and the metastable phases and it is shown to attract all the profiles which at ±∞ are in the domain of attraction of the stable and, respectively, the metastable states. The results are compared with those of Fife and Mc Leod for the Allen-Cahn equation, [13].