S Satheesh, E Sandhya
Methods of construction of Max-semi-selfdecompsable laws are given. Implications of this method in random time changed extremal processes are discussed. Max-autoregressive model is introduced and characterized using the max-semi-selfdecompsable laws and exponential max-semi-stable laws. Max-semi-selfddecomposability of max-semi-stable laws are proved.
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We discuss semi-selfdecomposable laws in the minimum scheme and characterize them using an autoregressive model. Semi-Pareto and semi-Weibull laws of Pillai (1991) are shown to be semi-selfdecomposable in this scheme. Methods for deriving this class of laws are then attempted from the angle of randomization. Finally, discrete analogues of these results are also considered.
The family of semi-stable laws is shown to be semi-selfdecomposable. Thus they qualify to model stationary first order autoregressive schemes. A connection between these autoregressive schemes with semi-stable marginals and semi-selfsimilar processes is given.
In this note we identify the class of distributions for {Xn} that can generate a linear, additive, first order auto-regressive scheme that is marginally stationary as semi-selfdecomposable laws. We give a method to construct these distributions. Its implications in subordination and selfdecomposability of Levy processes are given. The discrete analogues of these processes are also discussed.
It is shown that the hyperbolic functions can be associated with selfdecomposable distributions (in short: SD probability distributions or Lévy class L probability laws). Consequently, they admit associated background driving Lévy processes $Y$ (BDLP Y). We interpret the distributions of Y(1) via Bessel squared processes, Bessel bridges and local times.
We show that any freely selfdecomposable probability law is unimodal. This is the free probabilistic analog of Yamazato's result in [Ann. Probab. 6 (1978), 523-531].
For $\,0<α\le \infty$, new subclasses $\,\mathcal{U}^{<α>}$ of the class $\,\mathcal{U}$, of s-selfdecomposable probability measures, are studied. They are described by random integrals, by their characteristic functions and their Lévy spectral measures. Also their relations with the classical Lévy class $L$ of selfdecomposable distributions are investigated.
In this note we correct an omission in our paper (Satheesh and Sandhya, 2005) in defining semi-selfdecomposable laws and also show with examples that the marginal distributions of a stationary AR(1) process need not even be infinitely divisible.
Max-laws of large numbers for weakly dependent high dimensional arrays with applications
本文推导了高维弱依赖数组的最大定律大数定律，并应用于回归残差、相关筛选和高维参数测试。
We summarize the relations among three classes of laws: infinitely divisible, selfdecomposable and stable. First we look at them as the solutions of the Central Limit Problem; then their role is scrutinized in relation to the Levy and the additive processes with an emphasis on stationarity and selfsimilarity. Finally we analyze the Ornstein-Uhlenbeck processes driven by Levy noises and their selfdecomposable stationary distributions, and we end with a few particular examples.
Max stable laws are limit laws of linearly normalized partial maxima of indepen- dent, identically distributed (iid) random variables (rvs). These are analogous to stable laws which are limit laws of normalized partial sums of iid rvs. In this paper, we study entropy limit theorems for distribution functions in the max domain of attraction of max stable laws under linear normalization. More specifically, we study the problem of convergence of the Shannon entropy of linearly normalized partial maxima of iid rvs to the corresponding limit entropy when the linearly normalized partial maxima conve…
Continuing the study reported in Satheesh (2001),(arXiv:math.PR/0304499 dated 01May2003) here we study certain aspects of randomization in infinitely divisible (ID) and max-infinitely divisible (MID) laws. They generalize ID and MID laws. In particular we study mixtures of ID & MID laws, its relation to random sums & random maximums, corresponding stationary processes & extremal processes and some of their properties. It is shown that mixtures of ID laws and mixtures of MID laws appear as limits of random sums and random maximums respectively. We identify a class of probability generating func…
Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods
本文利用 Stein 方法、闭式技术和加权 Poincaré 型不等式，获得了自可分和对称 α-稳定概率测度的新稳定性估计，并计算了广义中心极限定理在 Wasserstein 型距离下的收敛速度，特别是 Cauchy 情形下的 $n^{-1}$ 收敛速度。
A transformation of gamma max-infinitely divisible laws viz. geometric gamma max-infinitely divisible laws is considered in this paper. Some of its distributional and divisibility properties are discussed and a random time changed extremal process corresponding to this distribution is presented. A new kind of invariance (stability) under geometric maxima is proved and a max-AR(1) model corresponding to it is also discussed.
Continuing the study reported in Satheesh (2001),(math.PR/0304499 dated 01 May 2003) and Satheesh (2002)(math.PR/0305030 dated 02May 2003), here we study generalizations of infinitely divisible (ID) and max-infinitely divisible (MID) laws. We show that these generalizations appear as limits of random sums and random maximums respectively. For the random sample size N, we identify a class of probability generating functions. Necessary and sufficient conditions that implies the convergence to an ID (MID) law by the convergence to these generalizations and vise versa are given. The results genera…
This paper proves that if a discrete distribution is infinitely divisible (ID) with integer-valued components, then it has a mass at the origin, which also implies why certain ID discrete laws do not have gaps in its support. We argue that discrete laws also can be stable and such laws do have domain of attraction. Then we give certain recent developments and references not reported in the Bose Dasgupta Rubin (2002) review in Sankhya, and some examples in the topics; infinite divisibility and stability of discrete laws, random infinite divisibility, operator stable laws, class-L laws, Goldie-S…
Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the $p$-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi \cite{chobanyan-l-s} for a sequence of random variables indexe…
Let $X$ be a max-stable random vector with positive continuous density. It is proved that the conditional independence of any collection of disjoint sub-vectors of $X$ given the remaining components implies their joint independence. We conclude that a broad class of tractable max-stable models cannot exhibit an interesting Markov structure.
The principal results of this contribution are the weak and strong limits of maxima of contracted stationary Gaussian random sequences. Due to the random contraction we introduce a modified Berman condition which is sufficient for the weak convergence of the maxima of the scaled sample. Under a stronger assumption the weak convergence is strengthened to almost convergence.
The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic processes and study the topology it induces on the space of laws of stochastic processes. This metric can be kernelized using the signature kernel which allows to efficiently compute it. As an application, we provide a non-parametric two-sample hypothesis test for laws of stochastic p…
Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative Lévy Processes
研究谱负Lévy过程在独立指数时间区间内最大回撤与最大回升的联合定律，通过路径分解结合Doob h变换及尺度函数推导其联合分布与矩的积分表示。
This paper develops Rio's method [C. R. Acad. Sci. Paris Sér. I Math., 1995] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.
We examine maximum vertex coloring of random geometric graphs, in an arbitrary but fixed dimension, with a constant number of colors. Since this problem is neither scale-invariant nor smooth, the usual methodology to obtain limit laws cannot be applied. We therefore leverage different concepts based on subadditivity to establish convergence laws for the maximum number of vertices that can be colored. For the constants that appear in these results, we provide the exact value in dimension one, and upper and lower bounds in higher dimensions.
Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(Ω,\mathscr{H},\widehat{\mathbb E})$, $S_n=X_1+\ldots+X_n$. We consider the moments of $\max_{n\ge 1}|S_n|/\sqrt{2n\log\log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.
Let $\{X_n\}_n$ be a sequence of freely independent, identically distributed non-commutative random variables. Consider a sequence $\{W_n\}_n$ of the renormalized spectral maximum of random variables $X_1,\cdots, X_n$. It is known that the renormalized spectral maximum $W_n$ converges to the free extreme value distribution under certain conditions on the distribution function. In this paper, we provide a rate of convergence in the Kolmogorov distance between a distribution function of $W_n$ and the free extreme value distribution.
Max-semistable extreme value laws for autoregressive processes with Cantor-like marginals
本文研究边缘分布类似康托尔函数的自动回归过程，证明其边缘分布属于极大半稳定分布的吸引域，并通过引入极值指数得到该过程的极值律，同时揭示其与确定性动力系统极值的联系。
Max-stable processes are widely used to model spatial extremes. These processes exhibit asymptotic dependence meaning that the large values of the process can occur simultaneously over space. Recently, inverted max-stable processes have been proposed as an important new class for spatial extremes which are in the domain of attraction of a spatially independent max-stable process but instead they cover the broad class of asymptotic independence. To study the extreme values of such processes we use the conditioned approach to multivariate extremes that characterises the limiting distribution of …
A general method to prove strong laws of large numbers for random fields is given. It is based on the Hájek - Rényi type method presented in Noszály and Tómács \cite{noszaly} and in Tómács and Líbor \cite{thomas06}. Noszály and Tómács \cite{noszaly} obtained a Hájek-Rényi type maximal inequality for random fields using moments inequalities. Recently, Tómács and Líbor \cite{thomas06} obtained a Hájek-Rényi type maximal inequality for random sequences based on probabilities, but not for random fields. In this paper we present a Hájek-Rényi type maximal inequality for random fields, using probabi…
The concept of a Lévy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the Lévy subordinator special case, the considered family is always strongly infinitely divisible with respect to time, meaning that a path can be represented in distribution as a finite sum with arbitrarily many summands of independent and identically distributed paths of another process. Besides distributional properties of the process, we present two applications to…
In this paper, by using the exact tail asymptotics derived by Debicki, Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the maximum of a class of non-homogeneous Gaussian random fields. By using the obtained results, we also derived the Gumbel laws for Shepp statistics of fractional Brownian motion and Gaussian integrated process as well as the Gumbel law for Storage process with fractional Brownian motion as input.
We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends to infinity, we consider the much more general setting in which the random points are the supports of appropriately defined Poisson processes. The main result covers the case of uniformly distributed points within a $d$-dimensional ellipsoid with a unique major axis. Moreover, several generalizations of the main result are established, for example a limit l…
We consider the spectrally negative Levy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Further, a new expression for the generalized Dickson's formula is provided.
Here we give a necessary and sufficient condition for the convergence to a random max infinitely divisible law from that of a random maximum. We then discuss random max-stable laws, their domain of max-attraction and the associated extremal processes.
We show how some attractive information--theoretic properties of Gaussians pass over to more general families of stable densities. We define a new score function for symmetric stable laws, and use it to give a stable version of the heat equation. Using this, we derive a version of the de Bruijn identity, allowing us to write the derivative of relative entropy as an inner product of score functions. We discuss maximum entropy properties of symmetric stable densities.
We attempt to bring some modest unity to three subareas of heavy tail analysis and extreme value theory: limit laws for componentwise maxima of iid random variables;hidden regular variation and asymptotic independence;conditioned limit laws when one component of a random vector is extreme. The common theme is multivariate regular variation on a cone and the three cases cited come from specifying the cones $[0,\infty]^d\setminus \{\boldsymbol 0\};(0,\infty]^d;$ and $[0,\infty]\times (0,\infty]$.
We consider stationary stochastic processes arising from dynamical systems by evaluating a given observable along the orbits of the system. We focus on the extremal behaviour of the process, which is related to the entrance in certain regions of the phase space, which correspond to neighbourhoods of the maximal set $\mathcal M$, i.e. the set of points where the observable is maximised. The main novelty here is the fact that we consider that the set $\mathcal M$ may have a countable number of points, which are associated by belonging to the orbit of a certain point, and may have accumulation po…
We study the max-convolution and max-stable laws for Boolean independence and prove that these are Dagum distributions (also known as log-logistical distributions).
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