Wave-length and amplitude for a stationary Gaussian process after a high maximum (original) (raw)

1972, Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete

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Abstract

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This paper investigates the distribution of wave-characteristics such as wave-length and amplitude in a stationary zero-mean Gaussian process, especially for high local maxima. Utilizing a conditioned random process, it analyzes the relationships between local maxima and subsequent minima, yielding ergodic distributions that describe the distances of these wave-characteristics.

On the gap and time interval between the first two maxima of long random walks

Journal of Statistical Mechanics: Theory and Experiment, 2014

In the context of order statistics of discrete time random walks (RW), we investigate the statistics of the gap, G n , and the number of time steps, L n , between the two highest positions of a Markovian one-dimensional random walker, starting from x 0 = 0, after n time steps (taking the x-axis vertical). The jumps η i = x i − x i−1 are independent and identically distributed random variables drawn from a symmetric probability distribution function (PDF), f (η), the Fourier transform of which has the small k behavior 1 −f (k) ∝ |k| µ , with 0 < µ ≤ 2. For µ = 2, the variance of the jump distribution is finite and the RW (properly scaled) converges to a Brownian motion. For 0 < µ < 2, the RW is a Lévy flight of index µ. We show that the joint PDF of G n and L n converges to a well defined stationary bi-variate distribution p(g, l) as the RW duration n goes to infinity. We present a thorough analytical study of the limiting joint distribution p(g, l), as well as of its associated marginals p gap (g) and p time (l), revealing a rich variety of behaviors depending on the tail of f (η) (from slow decreasing algebraic tail to fast decreasing super-exponential tail). We also address the problem for a random bridge where the RW starts and ends at the origin after n time steps. We show that in the large n limit, the PDF of G n and L n converges to the same stationary distribution p(g, l) as in the case of the free-end RW. Finally, we present a numerical check of our analytical predictions. Some of these results were announced in a recent letter [S. N. Majumdar,

Extremes of Gaussian Processes with Random Variance

Electronic Journal of Probability, 2011

Let ξ(t) be a standard locally stationary Gaussian process with covariance function 1 − r(t, t + s) ∼ C(t)|s| α as s → 0, with 0 < α ≤ 2 and C(t) a positive bounded continuous function. We are interested in the exceedance probabilities of ξ(t) with a random standard deviation η(t) = η − ζt β , where η and ζ are non-negative bounded random variables. We investigate the asymptotic behavior of the extreme values of the process ξ(t)η(t) under some specific conditions which depends on the relation between α and β.

Extreme statistics for time series: Distribution of the maximum relative to the initial value

Physical Review E, 2007

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/f^alpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRH_I). The exact MRH_I distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha, the distribution is determined from simulations. We find that the MRH_I distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRH_I distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some non-periodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRH_I distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

Abstracts for the Workshop on Extremes in Space and Time , May 31 On the extremal behavior of random variables observed at renewal times

2013

s for the Workshop on Extremes in Space and Time, May 31 On the extremal behavior of random variables observed at renewal times Bojan Basrak (University of Zagreb) This is joint work with Drago Špoljarić (University of Zagreb). We consider the asymptotic extremal behavior of iid observations X1, X2, . . ., until a random time τ(t) which is determined by a renewal process, possibly dependent on Xi’s. The maximum of these observations M(t) = max i≤τ(t) Xi , has been studied for decades. The first advances have already been made in the 1960s in the relatively straightforward case of a renewal process with finite mean interarrival times. Anderson [1] was the first to study the limiting behavior of M(t) in the case of a renewal process with infinite mean interarrival times. More recently, his result has been extended to describe the limiting behavior of (M(t)) at the level of processes (see [2], for instance). Using point processes techniques, we show how one can recover these known resu...

Asymptotic Poisson Character of Extremes in Non-Stationary Gaussian Models

Extremes, 2003

Let X be a non-stationary Gaussian process, asymptotically centered with constant variance. Let u be a positive real. Define R u (t) as the number of upcrossings of level u by the process X on the interval (0, t]. Under some conditions we prove that the sequence of point processes (R u ) u > 0 converges weakly, after normalization, to a standard Poisson process as u tends to infinity. In consequence of this study we obtain the weak convergence of the normalized supremum to a Gumbel distribution.

Extremes of independent Gaussian processes

Extremes, 2011

For every n ∈ N, let X 1n , . . . , Xnn be independent copies of a zero-mean Gaussian process Xn = {Xn(t), t ∈ T }. We describe all processes which can be obtained as limits, as n → ∞, of the process an(Mn − bn), where Mn(t) = max i=1,...,n X in (t) and an, bn are normalizing constants. We also provide an analogous characterization for the limits of the process anLn, where Ln(t) = min i=1,...,n |X in (t)|.

Certain bivariate distributions and random processes connected with maxima and minima

Extremes

The minimum and the maximum of t independent, identically distributed random variables haveF t and F t for their survival (minimum) and the distribution (maximum) functions, whereF = 1 − F and F are their common survival and distribution functions, respectively. We provide stochastic interpretation for these survival and distribution functions for the case when t > 0 is no longer an integer. A new bivariate model with these margins involve maxima and minima with a random number of terms. Our construction leads to a bivariate max-min process with t as its time argument. The second coordinate of the process resembles the well-known extremal process and shares with it the one-dimensional distribution given by F t. However, it is shown that the two processes are different. Some fundamental properties of the max-min process are presented, including a distributional Markovian characterization of its jumps and their locations.

On probability of high extremes for product of two independent Gaussian stationary processes

Extremes, 2014

Let X(t), Y (t), t ≥ 0, be two independent zero-mean stationary Gaussian processes, whose covariance functions are such that r i (t) = 1 − |t| a i + o(|t| a i) as t → 0, with 0 < a i ≤ 2, i = 1, 2 and both of the functions are less than one for non-zero t. We derive for any p the exact asymptotic behavior of the probability P (max t ∈[0,p] X(t)Y (t) > u) as u → ∞. We discuss possibilities generalizing obtained results to other Gaussian chaos processes h(X(t)), with a Gaussian vector process X(t) and a homogeneous function h of positive order.

On Clusters of High Extremes of Gaussian Stationary Processes with varepsilon\varepsilonvarepsilon-Separation

Electronic Journal of Probability, 2010

The clustering of extremes values of a stationary Gaussian process X (t), t ∈ [0, T ] is considered, where at least two time points of extreme values above a high threshold are separated by at least a small positive value ǫ. Under certain assumptions on the correlation function of the process, the asymptotic behavior of the probability of such a pattern of clusters of exceedances is derived exactly where the level to be exceeded by the extreme values, tends to ∞. The excursion behaviour of the paths in such an event is almost deterministic and does not depend on the high level u. We discuss the pattern and the asymptotic probabilities of such clusters of exceedances.

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References (14)

  1. Berman, S. M.: Limit theorems for the maximum term in stationary sequences. Ann. math. Statistics 35, 502-516 (1964).
  2. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968.
  3. Z.Wahrscheinlichkeitstheorie verw. Geb., Bd. 23
  4. Cram~r, H., Leadbetter, M. R.: Stationary and related stochastic processes. New York: Wiley 1967.
  5. Gupta, S. S.: Probability integrals of multivariate normal and multivariate t. Ann. math. Statistics 34, 792-828 (1963).
  6. Kac, M., Slepian, D.: Large excursions of Gaussian processes. Ann. math. Statistics 30, 1215-1228 (1959).
  7. Leadbetter, M. R., Weissner, E. W.: On continuity and other analytical properties of stochastic processes sample functions. Proc. Amer. math. Soc. 22, 291-294 (1969).
  8. Lindgren, G.: Some properties of a normal process near a local maximum. Ann. math. Statistics 41, 1870-1.883 (1970).
  9. Lindgren, G.: Extreme values of stationary normal processes. Z. Wahrscheinlichkeitstheorie verw. Geb. 17, 39-47 (1971).
  10. Lindgren, G.: Wave-length and amplitude in Gaussian noise. Advances appl. Probab. 4, 81-108 (1972).
  11. Quails, C.: On the joint distribution of crossings of high multiple levels by a stationary Gaussian process. Ark. Mat. 8, No. 15, 129-137 (1969).
  12. Quails, C., Watanabe, H.: An asymptotic 0-1 behavior of Gaussian processes. Ann. math. Statistics 42, 2029-2035 (1971).
  13. Slepian, D.: The one-sided barrier problem for Gaussian noise. Bell System techn. J. 41, 463-501 (1962). Georg Lindgren Department of Mathematical Statistics University of Lund Box 725 S-220 07 Lund 7
  14. Sweden (Received May 13, 1971 / in revised form February 28, 1972)

Some Properties of a Normal Process Near a Local Maximum

The Annals of Mathematical Statistics, 1970

Consider a stationary normal process Q(t) with mean zero and the covariance function r(t). Properties of the sample functions in the neighborhood of zeros, upcrossings of very high levels, etc. have been studied by, among others, Kac and Slepian, 1959 [4] and Slepian, 1962 [11]. In this paper we shall study the sample functions near local maxima of height u, especially as u-o, and mainly use similar methods as [4] and [11]. Then it is necessary to analyse carefully what is meant by "near a maximum of height u." In Section 2 we derive the "ergodic" definition, i.e. the definition which is possible to interpret by the aid of relative frequencies in a single realisation. 'lhis definition has been treated previously by Leadbetter, 1966 [5], and it turns out to be related to Kac and Slepian's horizontal window definition. In Section 3 we give a representation of {(t) near a maximum as the difference between a non-stationary normal process and a deterministic process, and in Section 4 we examine these processes as u-oo. We have then to distinguish between two cases. A: Regular case. r(t) = 1-A2 t2/2 + A4 t4/4!-A6 t6/6! + 0(t6) as t-O0, where the positive A2k are the spectral moments. Then it is proved that if Q(t) has a maximum of height u at t = 0 then, as u-+-oo, (Q24-2-2)1 {(24-42)-4-A22)t U-1) U} u-3{t4/4! + CO(A4-A22)A2-*t3/3!-04-A22)A2 1 t2/2} where co and C are independent random variables (rv), co has a standard normal distribution and C has the density z exp (-z), z > 0. Thus, in the neighborhood of a very low maximum the sample functions are fourth degree polynomials with positive t4-term, symmetrically distributed t3-term, and a negatively distributed t2-term but without t-term. B: Irregular case. r(t) = 1-A2t2/2+A4t4/4!-A% t15/5!+o(t5) as t-+ 0, where A5 > 0. Now ((tu-2)-U ,u 5{A2A5(A4-A22)-I tJ3/3! +(2A5)*0*(t)-(4-A22)A2 1t2/2} where co(t) is a non-stationary normal process whose second derivative is a Wiener process, independent of C which has the density zexp (-z), z > 0. The term A% t15/5! "disturbs" the process in such a way that the order of the distance which can be surveyed is reduced from I/lul (in Case A) to I/1u12. The results are used in Section 5 to examine the distribution of the wavelength and the crest-to-trough wave-height, i.e., the amplitude, discussed by, among

On the tails of the distribution of the maximum of a smooth stationary Gaussian process

ESAIM: Probability and Statistics, 2002

We study the tails of the distribution of the maximum of a stationary Gaussian process on a bounded interval of the real line. Under regularity conditions including the existence of the spectral moment of order 8, we give an additional term for this asymptotics. This widens the application of an expansion given originally by Piterbarg [11] for a sufficiently small interval.

Extremes of a certain class of Gaussian processes

Stochastic Processes and their Applications, 1999

We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend −ct ÿ for some constants c; ÿ ¿ 0 and a variance t 2H. We derive the tail behaviour of these extremes and show that they occur mainly in the neighbourhood of the unique point t0 where the related boundary function (u + ct ÿ)=t H is minimal. We consider the case that H ¡ ÿ.

Extremes of Gaussian Processes with Maximal Variance near the Boundary Points

Methodology And Computing In Applied Probability

Let Xt, t[0Y 1, be a Gaussian process with continuous paths with mean zero and nonconstant variance. The largest values of the Gaussian process occur in the neighborhood of the points of maximum variance. If there is a unique ®xed point t 0 in the interval 0Y 1, the behavior of Pfsup t[0Y1 Xt4ug is known for u??. We investigate the case where the unique point t 0 t u depends on u and tends to the boundary. This is reasonable for a family of Gaussian processes X u t depending on u, which have for each u such a unique point t u tending to the boundary as u??. We derive the asymptotic behavior of Pfsup t [ 0Y1 Xt4ug, depending on the rate as t u tends to 0 or 1. Some applications are mentioned and the computation of a particular case is used to compare simulated probabilities with the asymptotic formula. We consider the exceedances of such a nonconstant boundary by a Ornstein-Uhlenbeck process. It shows the dif®culties to simulate such rare events, when u is large.

Extremes of Gaussian processes with a smooth random variance

Stochastic Processes and their Applications, 2011

Let ξ(t) be a standard stationary Gaussian process with covariance function r (t), and η(t), another smooth random process. We consider the probabilities of exceedances of ξ(t)η(t) above a high level u occurring in an interval [0, T ] with T > 0. We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ(t)η(t), which is called the random variance process. We derive also a large deviation result for a general class of conditional Gaussian processes X (t) given a random element Y. c

On clusters of high extremes of Gaussian stationary processes with ǫ-separation

2010

The clustering of extremes values of a stationary Gaussian process X (t), t ∈ [0, T] is considered, where at least two time points of extreme values above a high threshold are separated by at least a small positive value ǫ. Under certain assumptions on the correlation function of the process, the asymptotic behavior of the probability of such a pattern of clusters of exceedances is derived exactly where the level to be exceeded by the extreme values, tends to∞. The excursion behaviour of the paths in such an event is almost deterministic and does not depend on the high level u. We discuss the pattern and the asymptotic probabilities of such clusters of exceedances.

Random Time-Changed Extremal Processes

Theory of Probability & Its Applications, 2007

The point process = {(T k , X k ) : k ≥ 1} we deal here with is assumed Bernoulli point process with independent random vectors X k in [0, ∞) d and with random time points T k in [0, ∞), independent of X. For normalizing we use a regular sequence ξ n (t, x) = (τ n (t), u n (x)) of timespace changes of [0, ∞) 1+d . We consider the sequence of the associated extremal processesỸ

International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo . June 28-30 , 2006 Extremes and ruin of Gaussian processes

2006

For certain Gaussian processes X(t) with trend −ct and variance V (t) we discuss maxima and ruin probabilities as well as the ruin time. The ruin time is defined as the first time point t such that X(t) − ct ≥ u where u stands typically for the initial capital. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications e.g. in telecommunications or storage models where it indicates the first time of an overflow. We deal with some asymptotic distributions of maxima, of ruin probability and of the ruin time as u → ∞. The limiting distributions are dependent on the parameters β, V (t) and the correlation function of X(t).

Cycle Range Distributions for Gaussian Processes Exact and Approximative Results

Extremes, 2004

Wave cycles, i.e., pairs of local maxima and minima, play an important role in many engineering fields. Many cycle definitions are used for specific purposes, such as crest-trough cycles in wave studies in ocean engineering and rainflow cycles for fatigue life predicition in mechanical engineering. The simplest cycle, that of a pair of local maximum and the following local minimum is also of interest as a basis for the study of more complicated cycles. This paper presents and illustrates modern computational tools for the analysis of different cycle distributions for stationary Gaussian processes with general spectrum. It is shown that numerically exact but slow methods will produce distributions in almost complete agreement with simulated data, but also that approximate and quick methods work well in most cases.