The two-dimensional Poisson process and extremal processes

1971 ◽  
Vol 8 (4) ◽  
pp. 745-756 ◽  
Author(s):  
James Pickands
Keyword(s):  
Probability Theory ◽  
Poisson Process ◽  
Order Statistics ◽  
Sample Space ◽  
Two Dimensional ◽  
Record Times ◽  
Extreme Order Statistics ◽  
Extremal Processes

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.

The two-dimensional Poisson process and extremal processes

1971 ◽  
Vol 8 (04) ◽  
pp. 745-756 ◽  
Author(s):  
James Pickands III
Keyword(s):  
Probability Theory ◽  
Poisson Process ◽  
Order Statistics ◽  
Sample Space ◽  
Two Dimensional ◽  
Record Times ◽  
Extreme Order Statistics ◽  
Extremal Processes

In recent years many applications of probability theory have involved such concepts as records, inter-record times and extreme order statistics. The results have generally been proved by diverse methods. In the present work a unifying structure is presented, which makes possible the simplification and extension of some of these results. The approach taken is to place all relevant processes on the same sample space. The underlying sample space is a homogeneous two-dimensional Poisson process.

Extremal processes and random measures

1975 ◽  
Vol 12 (2) ◽  
pp. 316-323 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Random Measure ◽  
Record Values ◽  
Poisson Processes ◽  
Two Dimensional ◽  
Random Measures ◽  
Record Times ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Gamma Processes

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

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.

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.

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

1975 ◽  
Vol 12 (3) ◽  
pp. 477-487 ◽  
Author(s):  
Ishay Weissman
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Independent Random Variables ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Extremal Processes

Let be the kth largest among Xn1, …, Xn[nt], where Xni = (Xi – an)/bn, {Xi} is a sequence of independent random variables and bn > 0 and an are norming constants. Suppose that for each converges in distribution. Then all the finite-dimensional laws of converge. The limiting process is represented in terms of a non-homogeneous two-dimensional 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.

Extremal processes and random measures

1975 ◽  
Vol 12 (02) ◽  
pp. 316-323
Author(s):  
R. W. Shorrock
Keyword(s):  
Random Measure ◽  
Record Values ◽  
Poisson Processes ◽  
Two Dimensional ◽  
Random Measures ◽  
Record Times ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Gamma Processes

Discrete time extremal processes with a continuous underlying c.d.f. are random measures which can be viewed as two-dimensional Poisson processes and this representation is used to obtain the conditional law of the sequence of states the process passes through (upper record values) given the sequence of holding times in states (inter-record times). In addition the Gamma processes (which lead to the Ferguson Dirichlet processes) and a random measure that arises in sampling from a biological population are discussed as two-dimensional Poisson processes.

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