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Almost Sure Continuity of Stable Moving Average Processes with Index Less Than One
Abstract
Rootzen (1978) gives a sufficient condition for sample continuity of moving average processes with respect to stable motion with index less than two. We provide a simple proof of this criterion for and show that the condition is then also necessary for continuity of the process. The same result holds for the moving-maximum process. A description of the local behaviour of the sample functions of such processes is given.
... It is easy to check directly that this process is stationary and simple maxstable. The almost sure continuity of the process follows from [1]. We think of t as a space parameter, not a time parameter. ...
... The a.s. continuity of the process follows from an extension of the arguments in [1]. The specific models we consider are analogous to the ones in the onedimensional situation: ...
... with a j,m := |t (a, b)) exp{− β 2 (a + b)}, which is decreasing in β for a, b > 0. Note thatβ (1) is simpler thanβ and summarizes the information of the sample in a somewhat more crude way. We could not find analogues ofβ (1) in two-dimensional space since we were unable to calculate explicitly the necessary higher-dimensional distributions. ...
The aim of this paper is to provide models for spatial extremes in the case of stationarity. The spatial dependence at extreme levels of a stationary process is modeled using an extension of the theory of max-stable processes of de Haan and Pickands [Probab. Theory Related Fields 72 (1986) 477--492]. We propose three one-dimensional and three two-dimensional models. These models depend on just one parameter or a few parameters that measure the strength of tail dependence as a function of the distance between locations. We also propose two estimators for this parameter and prove consistency under domain of attraction conditions and asymptotic normality under appropriate extra conditions.
... where ψ is a unimodal, continuous probability density function. While stationarity is obvious from Eq. (2), almost sure continuity follows from [1]. In this case ...
... A d-dimensional EVD G with standard negative exponential marginals can in general be represented as 1] is the Pickands dependence function of G [16]. ...
... The only difference is that β (1) is based on empirical quantiles as thresholds instead of theoretical ones, which does not require knowledge of the marginal distributions. The estimators β (1) andβ c,n should, therefore, be asymptotically equivalent in our setup. Actually, we show in Section 5 that the estimatorΨ c,n has asymptotically minimum variance within the class of regular estimators, if X follows a multivariate EVD or a multivariate GPD. ...
De Haan and Pereira (2006) [6] provided models for spatial extremes in the case of stationarity, which depend on just one parameter β>0 measuring tail dependence, and they proposed different estimators for this parameter. We supplement this framework by establishing local asymptotic normality (LAN) of a corresponding point process of exceedances above a high multivariate threshold. Standard arguments from LAN theory then provide the asymptotic minimum variance within the class of regular estimators of β. It turns out that the relative frequency of exceedances is a regular estimator sequence with asymptotic minimum variance, if the underlying observations follow a multivariate extreme value distribution or a multivariate generalized Pareto distribution.
... in the interesting field of infinite-dimensional extreme value theory, where the data are (continuous) functions. After the characterization of max-stable stochastic processes in C[0, 1] by Giné, Hahn and Vatan [11], de Haan and Lin [4, 5] investigated the domain of attraction conditions and established weak consistency of estimators of the extreme value index, the centering and standardizing sequences, and the exponent measure. Statistics of infinite-dimensional extremes will find various applications, for example, to coast protection (flooding) and risk assessment in finance. ...
... In finance, the intra-day return of a stock is defined as the ratio of the price of a stock at a certain time t during the day to the price at market opening. This process can be well described, when we measure time in days, with a continuous function on [0, 1]. For various risk analysis problems (e.g., problems dealing with options), intra-day returns of the stock need to be taken into account, instead of just the daily returns (i.e., the function values at 1). ...
... Sampling on n days puts us in a position to apply statistics of extremes to these problems. Also, from a mathematical point of view, the research is challenging, because of the new features of C[0, 1]-valued random elements, when compared to random variables or vectors, in particular, the uniformity in t ∈ [0, 1] of the results asks for novel approaches. It is the purpose of this paper to establish asymptotic normality of estimators of the extreme value index, which is now an element of C[0, 1], and of the normalizing sequences. ...
Consider n i.i.d. random elements on C[0; 1]. We show that under an appropriate strengthening of the domain of attraction condition natural estimators of the extreme-value index, which is now a continuous function, and the normalizing functions have a Gaussian process as limiting distribution. A key tool is the weak convergence of a weighted tail empirical process, which makes it possible to obtain the results uniformly on [0; 1]. Detailed examples are also presented.
... It is easy to check directly that this process is stationary and simple maxstable. The almost sure continuity of the process follows from [1]. We think of t as a space parameter, not a time parameter. ...
... The a.s. continuity of the process follows from an extension of the arguments in [1]. The specific models we consider are analogous to the ones in the onedimensional situation: ...
... with a j,m := |t (a, b)) exp{− β 2 (a + b)}, which is decreasing in β for a, b > 0. Note thatβ (1) is simpler thanβ and summarizes the information of the sample in a somewhat more crude way. We could not find analogues ofβ (1) in two-dimensional space since we were unable to calculate explicitly the necessary higher-dimensional distributions. ...
The aim of this paper is to provide models for spatial extremes in the case of stationarity. The spatial dependence at extreme levels of a stationary process is modeled using an extension of the theory of max-stable processes of de Haan and Pickands [Probab. Theory Related Fields 72 (1986) 477--492]. We propose three one-dimensional and three two-dimensional models. These models depend on just one parameter or a few parameters that measure the strength of tail dependence as a function of the distance between locations. We also propose two estimators for this parameter and prove consistency under domain of attraction conditions and asymptotic normality under appropriate extra conditions.
... However, the results , which are often quite sharp in determining whether (1.1) has a continuous version, are not useful in determining whether it has a version which is bounded, except in some special cases. (See also the related work of Rootzen [10], Balkema and De Haan [1] on the continuity of stable moving averages) . In this note we approach this question in a completely different, and much more elementary way, and obtain some simple conditions for (1.1) to be bounded almost surely. ...
... Here we use Khintchine's inequality, see e.g. [2], and the same repre Note that when J is continuous then so is [1] (since J(t) = 0 for t < 0, the assumption J(O) = 0 is needed here). Consequently, Y3 given by (3 .2) is continuous. ...
We show that the moving average process \begin{array}{*{20}{c}} {{{\chi }_{f}}(t): = \int_{0}^{t} {f(t - s)dZ(s)} } & {t \in [0,T]} \\ \end{array}
has a bounded version almost surely, when the kernel f has finite total 2-variation and Z is a symmetric Lévy process. We also obtain bounds for and for uniform moduli of continuity of and for the largest jump of when it is not continuous. Similar results are obtained for forward average processes. The methods developed are also used to show that certain infinitely divisible random fields are bounded.
In this paper, a P‐min‐stable regression model is proposed for a time series of extreme values observed in a limited interval. The model may be useful when the variable or indicator of interest is the minimum value of a series restricted to the unit interval and is related to other variables through a regression structure. The serial extremal dependence is induced through the marginalization of the Kumaraswamy distribution conditioned on a latent ‐stable process. The model is flexible to capture trends, seasonality, and non‐stationarity. Some properties of the model are presented, as well as the extremogram of the series. Procedures for estimation and inference are discussed and implemented via an Expectation‐Maximization algorithm. As an illustration, the model was used to analyze the minimum relative humidity observed in the Brazilian Amazon.
In the previous chapter we provided the probabilistic background for general regularly varying strictly stationary processes \d ({\mathbf {X}}_t)_{t\in {\mathbb Z}} with generic element . In this chapter we consider various important classes of time series models which have the regular variation property. We focus on the derivation of the corresponding tail measures and the spectral tail process. In the presence of serial dependence the tail measures and spectral tail processes often have some interesting structure. They characterize the appearance of extremal dependence clusters and the propagation of an extreme event at a given time to future observations.
Extreme Value Theory (EVT) is a branch of statistics that focuses on the study of rare events or extreme values. It has applications in various domains like finance (e.g., modeling extreme losses or gains), meteorology (e.g., modeling extreme rainfall or temperature events), and environmental sciences (e.g., modeling extreme flood levels). EVT provides tools and methodologies to estimate the probability of extreme events and to model their behavior.
The expected number of upcrossings for a max-stable process is computed and compared with known results for stable processes. Asymptotically the formulas are of the same order.
We review recent developments in the theory of extremal processes, and their applications to modeling the choice behavior of individuals who behave according to the principle of random utility maximization. We trace the evolution of the modeling of the choice behavior of individuals beginning with intertemporal choice behavior where the set of alternatives is finite. Then we discuss models of continuous choice, where the choice set is no longer finite but an arbitrary compact set. The latter models are developed with max-stable processes representing the utility functions at any fixed point in time. We describe dynamic extensions of the modeling of continuous choice, which require extensions of the finite dimensional theory of classical extremal processes to infinite dimensions.
The expected number of upcrossings for a max-stable process is computed and compared with known results for stable processes. Asymptotically the formulas are of the same order.
A max-autoregressive moving average (MARMA(p,q)) process {Xt} satisfies the recursion Xt=φ 1Xt-1∨ ⋯ ∨ φ pXt-p∨ Zt∨ θ 1Zt-1∨ ⋯ ∨ θ qZt-q for all t where φ i, θ j≥ 0, and {Zt} is i.i.d. with common distribution function for . Such processes have finite-dimensional distributions which are max-stable and hence are examples of max-stable processes. We provide necessary and sufficient conditions for existence of a stationary solution to the MARMA recursion and we examine the reducibility of the process to a MARMA(p′,q′) with $p^{\prime}
Since the publication of his masterpiece on regular variation and its application to the weak convergence of (univariate)
sample extremes in 1970, Laurens de Haan (Thesis, Mathematical Centre Tract vol. 32, University of Amsterdam, 1970) is among the leading mathematicians in the world, with a particular focus on extreme value theory (EVT). On the occasion
of his 70th birthday it is a great pleasure and a privilege to follow his route through multivariate EVT, which started only seven years later in 1977, when Laurens de Haan published his first paper on multivariate EVT, jointly
with Sid Resnick.
We discuss moving-maximum models, based on weighted maxima of independent random variables, for extreme values from a time series. The models encompass a range of stochastic processes that are of interest in the context of extreme-value data. We show that a stationary stochastic process whose finite-dimensional distributions are extreme-value distributions may be approximated arbitrarily closely by a moving-maximum process with extreme-value marginals. It is demonstrated that bootstrap techniques, applied to moving-maximum models, may be used to construct confidence and prediction intervals from dependent extrema. Moreover, it is shown that bootstrapped moving-maximum models may be used to capture the dominant features of a range of processes that are not themselves moving maxima. Connections of moving-maximum models to more conventional, moving-average processes are addressed. In particular, it is proved that a moving-maximum process with extreme-value distributed marginals may be approximated by powers of moving-average processes with stably distributed marginals.
The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into Lp (0 < p ≤ 2). We use the theory of Lp isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex Lq into real or complex Lp for 0 < p < q ≤ 2.
In this paper we study extremes of non-normal stable moving average processes, i.e., of stochastic processes of the form or , where is stable with index . The extremes are described as a marked point process, consisting of the point process of (separated) exceedances of a level together with marks associated with the points, a mark being the normalized sample path of X(t) around an exceedance. It is proved that this marked point process converges in distribution as the level increases to infinity. The limiting distribution is that of a Poisson process with independent marks which have random heights but otherwise are deterministic. As a byproduct of the proof for the continuous-time case, a result on sample path continuity of stable processes is obtained.