Non-homogeneously paced random records and associated extremal processes

1978 ◽  
Vol 15 (03) ◽  
pp. 552-559 ◽  
Author(s):  
Donald P. Gaver ◽  
Patricia A. Jacobs
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rate Function ◽  
Extreme Value ◽  
Value Distribution ◽  
Homogeneous Poisson Process ◽  
Extremal Process ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

Non-homogeneously paced random records and associated extremal processes

1978 ◽  
Vol 15 (3) ◽  
pp. 552-559 ◽  
Author(s):  
Donald P. Gaver ◽  
Patricia A. Jacobs
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rate Function ◽  
Extreme Value ◽  
Value Distribution ◽  
Homogeneous Poisson Process ◽  
Extremal Process ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

Extremal processes and record value times

1973 ◽  
Vol 10 (4) ◽  
pp. 864-868 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Poisson Process ◽  
Homogeneous Poisson Process ◽  
Record Times ◽  
Limiting Behavior ◽  
Extremal Process ◽  
Record Value ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

Let {Xn, n ≧ 1} be i.i.d. and Yn = max {X1,…, Xn}. Xj is a record value of {Xn} if Yj > Yj–1 The record value times are Ln, n ≧ 1 and inter-record times are Δn, n ≧ 1. The known limiting behavior of {Ln} and {Δn} is close to that of a non-homogeneous Poisson process and an explanation of this is obtained by embedding {Yn} in a suitable extremal process which jumps according to a non-homogeneous Poisson process.

Extremal processes and record value times

1973 ◽  
Vol 10 (04) ◽  
pp. 864-868 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Poisson Process ◽  
Homogeneous Poisson Process ◽  
Record Times ◽  
Limiting Behavior ◽  
Extremal Process ◽  
Record Value ◽  
Extremal Processes ◽  
Non Homogeneous Poisson Process

Let {Xn , n ≧ 1} be i.i.d. and Yn = max {X 1,…, Xn }. Xj is a record value of {Xn } if Yj > Yj– 1 The record value times are Ln, n ≧ 1 and inter-record times are Δ n , n ≧ 1. The known limiting behavior of {Ln } and {Δn } is close to that of a non-homogeneous Poisson process and an explanation of this is obtained by embedding {Yn } in a suitable extremal process which jumps according to a non-homogeneous Poisson process.

scholarly journals Tail Properties and Asymptotic Expansions for the Maximum of the Logarithmic Skew-Normal Distribution

2013 ◽  
Vol 50 (3) ◽  
pp. 900-907 ◽  
Author(s):  
Xin Liao ◽  
Zuoxiang Peng ◽  
Saralees Nadarajah
Keyword(s):  
Asymptotic Expansion ◽  
Convergence Rate ◽  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Skew Normal Distribution ◽  
Skew Normal

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

Limit laws for maxima of a sequence of random variables defined on a Markov chain

1970 ◽  
Vol 2 (2) ◽  
pp. 323-343 ◽  
Author(s):  
Sidney I. Resnick ◽  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Distributions ◽  
Limit Laws ◽  
Limiting Behavior ◽  
Normalizing Constants

Consider the bivariate sequence of r.v.'s {(Jn, Xn), n ≧ 0} with X0 = - ∞ a.s. The marginal sequence {Jn} is an irreducible, aperiodic, m-state M.C., m < ∞, and the r.v.'s Xn are conditionally independent given {Jn}. Furthermore P{Jn = j, Xn ≦ x | Jn − 1 = i} = pijHi(x) = Qij(x), where H1(·), · · ·, Hm(·) are c.d.f.'s. Setting Mn = max {X1, · · ·, Xn}, we obtain P{Jn = j, Mn ≦ x | J0 = i} = [Qn(x)]i, j, where Q(x) = {Qij(x)}. The limiting behavior of this probability and the possible limit laws for Mn are characterized.Theorem. Let ρ(x) be the Perron-Frobenius eigenvalue of Q(x) for real x; then:(a)ρ(x) is a c.d.f.;(b) if for a suitable normalization {Qijn(aijnx + bijn)} converges completely to a matrix {Uij(x)} whose entries are non-degenerate distributions then Uij(x) = πjρU(x), where πj = limn → ∞pijn and ρU(x) is an extreme value distribution;(c) the normalizing constants need not depend on i, j;(d) ρn(anx + bn) converges completely to ρU(x);(e) the maximum Mn has a non-trivial limit law ρU(x) iff Qn(x) has a non-trivial limit matrix U(x) = {Uij(x)} = {πjρU(x)} or equivalently iff ρ(x) or the c.d.f. πi = 1mHiπi(x) is in the domain of attraction of one of the extreme value distributions. Hence the only possible limit laws for {Mn} are the extreme value distributions which generalize the results of Gnedenko for the i.i.d. case.

scholarly journals Volcanic hazard assessment for the Canary Islands (Spain) using extreme value theory

2011 ◽  
Vol 11 (10) ◽  
pp. 2741-2753 ◽  
Author(s):  
R. Sobradelo ◽  
J. Martí ◽  
A. T. Mendoza-Rosas ◽  
G. Gómez
Keyword(s):  
Time Series ◽  
Poisson Process ◽  
Statistical Method ◽  
Canary Islands ◽  
Extreme Value Theory ◽  
Volcanic Hazard ◽  
Value Theory ◽  
Extreme Value ◽  
Homogeneous Poisson Process ◽  
Non Homogeneous Poisson Process

Abstract. The Canary Islands are an active volcanic region densely populated and visited by several millions of tourists every year. Nearly twenty eruptions have been reported through written chronicles in the last 600 yr, suggesting that the probability of a new eruption in the near future is far from zero. This shows the importance of assessing and monitoring the volcanic hazard of the region in order to reduce and manage its potential volcanic risk, and ultimately contribute to the design of appropriate preparedness plans. Hence, the probabilistic analysis of the volcanic eruption time series for the Canary Islands is an essential step for the assessment of volcanic hazard and risk in the area. Such a series describes complex processes involving different types of eruptions over different time scales. Here we propose a statistical method for calculating the probabilities of future eruptions which is most appropriate given the nature of the documented historical eruptive data. We first characterize the eruptions by their magnitudes, and then carry out a preliminary analysis of the data to establish the requirements for the statistical method. Past studies in eruptive time series used conventional statistics and treated the series as an homogeneous process. In this paper, we will use a method that accounts for the time-dependence of the series and includes rare or extreme events, in the form of few data of large eruptions, since these data require special methods of analysis. Hence, we will use a statistical method from extreme value theory. In particular, we will apply a non-homogeneous Poisson process to the historical eruptive data of the Canary Islands to estimate the probability of having at least one volcanic event of a magnitude greater than one in the upcoming years. This is done in three steps: First, we analyze the historical eruptive series to assess independence and homogeneity of the process. Second, we perform a Weibull analysis of the distribution of repose time between successive eruptions. Third, we analyze the non-homogeneous Poisson process with a generalized Pareto distribution as the intensity function.

The penultimate form of approximation to normal extremes

1982 ◽  
Vol 14 (02) ◽  
pp. 324-339 ◽  
Author(s):  
Jonathan P. Cohen
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Value Distribution ◽  
Gaussian Sequence ◽  
Stationary Gaussian Sequence ◽  
Type Iii

Let Yn denote the largest of n independent N(0,1) random variables. It is shown that the error in approximating the distribution of Yn by the type III extreme value distribution exp {– (–Ax + B) k }, k &gt; 0, is uniformly of order (log n)–2 if and only if the constants A, B and k satisfy certain conditions. In particular, this holds for the penultimate form of Fisher and Tippett (1928). Furthermore, two sufficient conditions are given so that these results can be extended to a stationary Gaussian sequence.

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