Statistical Inference for Max-Stable Processes by Conditioning on Extreme Events

2014 ◽  
Vol 46 (02) ◽  
pp. 478-495 ◽  
Author(s):  
Sebastian Engelke ◽  
Alexander Malinowski ◽  
Marco Oesting ◽  
Martin Schlather
Keyword(s):  
Stable Process ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Shape Functions ◽  
Peaks Over Threshold ◽  
New Methods ◽  
Convergence Results ◽  
Block Maxima ◽  
Threshold Approach ◽  
Mixed Moving Maxima

In this paper we provide the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.

Statistical Inference for Max-Stable Processes by Conditioning on Extreme Events

2014 ◽  
Vol 46 (2) ◽  
pp. 478-495 ◽  
Author(s):  
Sebastian Engelke ◽  
Alexander Malinowski ◽  
Marco Oesting ◽  
Martin Schlather
Keyword(s):  
Stable Process ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Shape Functions ◽  
Peaks Over Threshold ◽  
New Methods ◽  
Convergence Results ◽  
Block Maxima ◽  
Threshold Approach ◽  
Mixed Moving Maxima

In this paper we provide the basis for new methods of inference for max-stable processes ξ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process η in the max-domain of attraction of ξ on being extremal, several convergence results for the increments of η are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events η. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.

scholarly journals A generalization of Matérn hard-core processes with applications to max-stable processes

2020 ◽  
Vol 57 (4) ◽  
pp. 1298-1312
Author(s):  
Martin Dirrler ◽  
Christopher Dörr ◽  
Martin Schlather
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Hard Core ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Poisson Point Processes ◽  
Original Process ◽  
Mixed Moving Maxima

AbstractMatérn hard-core processes are classical examples for point processes obtained by dependent thinning of (marked) Poisson point processes. We present a generalization of the Matérn models which encompasses recent extensions of the original Matérn hard-core processes. It generalizes the underlying point process, the thinning rule, and the marks attached to the original process. Based on our model, we introduce processes with a clear interpretation in the context of max-stable processes. In particular, we prove that one of these processes lies in the max-domain of attraction of a mixed moving maxima process.

Some Functional Stable Limit Theorems

1973 ◽  
Vol 16 (2) ◽  
pp. 173-177 ◽  
Author(s):  
D. R. Beuerman
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Stable Process ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Limit ◽  
Stable Law ◽  
Partial Sum Process ◽  
Image Position

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,1whereandwhere L(n) is a function of slow variation; also take S0=0, B0=l.In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

Characterization of stable processes by identically distributed stochastic integrals

1980 ◽  
Vol 12 (3) ◽  
pp. 689-709 ◽  
Author(s):  
M. Riedel
Keyword(s):  
Stochastic Process ◽  
Stable Process ◽  
Convergence In Probability ◽  
Stochastic Integrals ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Subject

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

scholarly journals Ruin probability with certain stationary stable claims generated by conservative flows

2007 ◽  
Vol 39 (02) ◽  
pp. 360-384 ◽  
Author(s):  
Uğur Tuncay Alparslan ◽  
Gennady Samorodnitsky
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Ruin Probability ◽  
Stable Process ◽  
Independent Interest ◽  
Stable Processes ◽  
Auxiliary Result ◽  
Initial Capital ◽  
Order Of Magnitude

We study the ruin probability where the claim sizes are modeled by a stationary ergodic symmetric α-stable process. We exploit the flow representation of such processes, and we consider the processes generated by conservative flows. We focus on two classes of conservative α-stable processes (one discrete-time and one continuous-time), and give results for the order of magnitude of the ruin probability as the initial capital goes to infinity. We also prove a solidarity property for null-recurrent Markov chains as an auxiliary result, which might be of independent interest.

On dispersion of stable random vectors and its application in the prediction of multivariate stable processes

1994 ◽  
Vol 31 (3) ◽  
pp. 691-699 ◽  
Author(s):  
A. Reza Soltani ◽  
R. Moeanaddin
Keyword(s):  
Stable Process ◽  
Stable Processes ◽  
Random Vectors ◽  
Linear Predictor ◽  
Error Vector ◽  
Best Linear Predictor ◽  
Definition Of ◽  
Stable Random Vectors

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric α stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.

On the maximum of sums of random variables and the supremum functional for stable processes

1969 ◽  
Vol 6 (2) ◽  
pp. 419-429 ◽  
Author(s):  
C.C. Heyde
Keyword(s):  
Limit Theorem ◽  
Stable Distribution ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Law ◽  
Sums Of Random Variables ◽  
Symmetric Stable Distribution ◽  
General Limit ◽  
General Limit Theorem

Let Xi, i = 1, 2, 3, … be a sequence of independent and identically distributed random variables which belong to the domain of attraction of a stable law of index a. Write S0= 0, Sn = Σ i=1nXi, n ≧ 1, and Mn = max0 ≦ k ≦ nSk. In the case where the Xi are such that Σ1∞n−1Pr(Sn > 0) < ∞, we have limn→∞Mn = M which is finite with probability one, while in the case where Σ1∞n−1Pr(Sn < 0) < ∞, a limit theorem for Mn has been obtained by Heyde [9]. The techniques used in [9], however, break down in the case Σ1∞n−1Pr(Sn < 0) < ∞, Σ1∞n−1Pr(Sn > 0) < ∞ (the case of oscillation of the random walk generated by the Sn) and the only results available deal with the case α = 2 (Erdos and Kac [5]) and the case where the Xi themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit theorem for Mn in the case of oscillation.

scholarly journals Exceedance of power barriers for integrated continuous-time stationary ergodic stable processes

2009 ◽  
Vol 41 (03) ◽  
pp. 874-892
Author(s):  
Uğur Tuncay Alparslan
Keyword(s):  
Continuous Time ◽  
Stable Process ◽  
Tail Probability ◽  
Exceedance Probability ◽  
Dependence Structure ◽  
Stable Processes ◽  
Limiting Behavior ◽  
Order Of Magnitude ◽  
The Difference ◽  
Conservative Flow

We study the asymptotic behavior of the tail probability of integrated stable processes exceeding power barriers. In the first part of the paper the limiting behavior of the integrals of stable processes generated by ergodic dissipative flows is established. In the second part an example with the integral of a stable process generated by a conservative flow is analyzed. Finally, the difference in the order of magnitude of the exceedance probability in the two cases is related to the dependence structure of the underlying stable process.

Hilbert Spaces Formed by Strongly Harmonizable Stable Processes

2001 ◽  
Vol 8 (1) ◽  
pp. 181-188
Author(s):  
A. R. Soltani ◽  
B. Tarami
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Continuous Time ◽  
Moving Average ◽  
Stable Process ◽  
Stable Processes ◽  
Moving Average Representation ◽  
Process Elements ◽  
Average Representation

Abstract A strongly harmonizable continuous time symmetric α-stable process is considered. By using covariations, a Hilbert space is formed from the process elements and used for a purpose of moving average representation and prediction.

Sign in / Sign up

Export Citation Format

Share Document