Extremal processes, secretary problems and the 1/e law

10.2307/3214377 ◽

1989 ◽

Vol 26 (4) ◽

pp. 722-733 ◽

Cited By ~ 28

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.

Inverses of extremal processes

Advances in Applied Probability ◽

10.2307/1426300 ◽

1974 ◽

Vol 6 (2) ◽

pp. 392-406 ◽

Cited By ~ 22

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

Advances in Applied Probability ◽

10.1017/s0001867800045432 ◽

1974 ◽

Vol 6 (02) ◽

pp. 392-406 ◽

Cited By ~ 7

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.

Small-time almost-sure behaviour of extremal processes

Advances in Applied Probability ◽

10.1017/apr.2017.7 ◽

2017 ◽

Vol 49 (2) ◽

pp. 411-429 ◽

Cited By ~ 1

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

10.2307/3213118 ◽

1978 ◽

Vol 15 (3) ◽

pp. 552-559 ◽

Cited By ~ 8

Author(s):

Donald P. Gaver ◽

Patricia A. Jacobs

Keyword(s):

Poisson Process ◽

Random Variables ◽

Extreme Value Distribution ◽

Rate Function ◽

Extreme Value ◽

Value Distribution ◽

Extremal Process ◽

Extremal Processes ◽

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

10.2307/3212388 ◽

1973 ◽

Vol 10 (4) ◽

pp. 864-868 ◽

Cited By ~ 29

Author(s):

Sidney I. Resnick

Keyword(s):

Poisson Process ◽

Record Times ◽

Limiting Behavior ◽

Extremal Process ◽

Record Value ◽

Extremal Processes ◽

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.

Non-homogeneously paced random records and associated extremal processes

10.1017/s0021900200045927 ◽

1978 ◽

Vol 15 (03) ◽

pp. 552-559 ◽

Cited By ~ 2

Author(s):

Donald P. Gaver ◽

Patricia A. Jacobs

Keyword(s):

Poisson Process ◽

Random Variables ◽

Extreme Value Distribution ◽

Rate Function ◽

Extreme Value ◽

Value Distribution ◽

Extremal Process ◽

Extremal Processes ◽

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

10.1017/s0021900200096054 ◽

1973 ◽

Vol 10 (04) ◽

pp. 864-868 ◽

Cited By ~ 10

Author(s):

Sidney I. Resnick

Keyword(s):

Poisson Process ◽

Record Times ◽

Limiting Behavior ◽

Extremal Process ◽

Record Value ◽

Extremal Processes ◽

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.