Inverses of extremal processes

1974 ◽  
Vol 6 (2) ◽  
pp. 392-406 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Random Variables ◽  
Lévy Measure ◽  
Additive Process ◽  
Extremal Process ◽  
Simple Transformation ◽  
Extremal Processes

The inverse of an extremal process {Y(t), t ≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps of Y while Y is in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{– e–x} is also extremal-Λ (x) and this fact enables one to relate behavior of Y-Λ at t = ∞ to behavior near t = 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.

Inverses of extremal processes

1974 ◽  
Vol 6 (02) ◽  
pp. 392-406 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Random Variables ◽  
Lévy Measure ◽  
Additive Process ◽  
Extremal Process ◽  
Simple Transformation ◽  
Extremal Processes

The inverse of an extremal process {Y(t),t≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps ofYwhileYis in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{–e–x} is also extremal-Λ (x) and this fact enables one to relate behavior ofY-Λ att= ∞ to behavior neart= 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.

scholarly journals Small-time almost-sure behaviour of extremal processes

2017 ◽  
Vol 49 (2) ◽  
pp. 411-429 ◽  
Author(s):  
Ross A. Maller ◽  
Peter C. Schmidli
Keyword(s):  
Continuous Time ◽  
Relative Stability ◽  
Random Variables ◽  
Nondecreasing Function ◽  
Small Time ◽  
Time Behaviour ◽  
Integral Criterion ◽  
Extremal Process ◽  
Extremal Processes ◽  
Maximum Sequence

Abstract An rth-order extremal process Δ(r) = (Δ(r)t)t≥0 is a continuous-time analogue of the rth partial maximum sequence of a sequence of independent and identically distributed random variables. Studying maxima in continuous time gives rise to the notion of limiting properties of Δt(r) as t ↓ 0. Here we describe aspects of the small-time behaviour of Δ(r) by characterising its upper and lower classes relative to a nonstochastic nondecreasing function bt > 0 with limt↓bt = 0. We are then able to give an integral criterion for the almost sure relative stability of Δt(r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δt(1) as t ↓ 0.

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.

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.

Embedding sequences of successive maxima in extremal processes, with applications

1987 ◽  
Vol 24 (4) ◽  
pp. 827-837 ◽  
Author(s):  
Rocco Ballerini ◽  
Sidney I. Resnick
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Random Variables ◽  
Record Values ◽  
Record Times ◽  
Extremal Processes ◽  
Sample Record

Consequences of embedding sequences {Mn} in an extremal-F process are discussed where Mn represents the maximum of n independent (but not necessarily identically distributed) random variables. Various limit theorems are proved for the sample record rate, record times, inter-record times, and record values. These results are illustrated with applications to three particular record models: the Yang (1975) record model where population size increases geometrically, a record model where linear improvement is present, and a record model incorporating features of the previous two.

Embedding sequences of successive maxima in extremal processes, with applications

1987 ◽  
Vol 24 (04) ◽  
pp. 827-837 ◽  
Author(s):  
Rocco Ballerini ◽  
Sidney I. Resnick
Keyword(s):  
Population Size ◽  
Limit Theorems ◽  
Random Variables ◽  
Record Values ◽  
Record Times ◽  
Extremal Processes ◽  
Sample Record

Consequences of embedding sequences {Mn } in an extremal-F process are discussed where Mn represents the maximum of n independent (but not necessarily identically distributed) random variables. Various limit theorems are proved for the sample record rate, record times, inter-record times, and record values. These results are illustrated with applications to three particular record models: the Yang (1975) record model where population size increases geometrically, a record model where linear improvement is present, and a record model incorporating features of the previous two.

Multivariate extremal processes generated by independent non-identically distributed random variables

1975 ◽  
Vol 12 (03) ◽  
pp. 477-487 ◽  
Author(s):  
Ishay Weissman
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Independent Random Variables ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Extremal Processes

Letbe thekth largest amongXn1, …,Xn[nt], whereXni= (Xi– an)/bn, {Xi} is a sequence of independent random variables andbn> 0 andanare norming constants. Suppose that for eachconverges in distribution. Then all the finite-dimensional laws ofconverge. The limiting process is represented in terms of a non-homogeneous two-dimensional Poisson process.

scholarly journals On regular variation for infinitely divisible random vectors and additive processes

2006 ◽  
Vol 38 (01) ◽  
pp. 134-148 ◽  
Author(s):  
Henrik Hult ◽  
Filip Lindskog
Keyword(s):  
Stochastic Processes ◽  
Regular Variation ◽  
Tail Behavior ◽  
Random Vectors ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Additive Process ◽  
Asymptotic Decay ◽  
Stationary Increments ◽  
Additive Processes

We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

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, secretary problems and the 1/e law

1989 ◽  
Vol 26 (04) ◽  
pp. 722-733 ◽  
Author(s):  
Dietmar Pfeifer
Keyword(s):  
Extremal Process ◽  
Winning Probability ◽  
Extremal Processes

We consider a class of secretary problems in which the order of arrival of candidates is no longer uniformly distributed. By a suitable embedding in a time-transformed extremal process it is shown that the asymptotic winning probability is again 1/e as in the classical situation. Extensions of the problem to more than one choice are also considered.

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