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.

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.

Records in the presence of a linear trend

1987 ◽  
Vol 19 (04) ◽  
pp. 801-828 ◽  
Author(s):  
Rocco Ballerini ◽  
Sidney I. Resnick
Keyword(s):  
Gaussian Process ◽  
Limit Theorems ◽  
Linear Trend ◽  
Random Sequence ◽  
Record Values ◽  
Stationary Gaussian Process ◽  
Record Times

Records from the sequence Yn = Xn + cn, n ≧ 1 are analyzed, where {Xn } is a strictly stationary random sequence. We prove various limit theorems for the record rate, record times, and record values. The situation when {Xn } is a stationary Gaussian process is considered with special attention given to Gaussian ARMA sequences. Data for the 400 and 800 metre races are used to illustrate these results.

On discrete time extremal processes

1974 ◽  
Vol 6 (3) ◽  
pp. 580-592 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Population Growth ◽  
Discrete Time ◽  
Continuous Time ◽  
Record Values ◽  
Record Times ◽  
State Spaces ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Continuous Time Processes

Upper record values and times and inter-record times are studied in their rôles as embedded structures in discrete time extremal processes. Various continuous time approximations to the discrete-time processes are analysed, especially as processes over their state spaces. Discrete time processes, suitably normalized after crossing a threshold T, are shown to converge to limiting continuous time processes as T → ∞ under suitable assumptions on the underlying CDF F, for example, when 1 — F varies regularly at ∞, and more generally. Discrete time extremal processes viewed as processes over their state spaces are noted to have an interesting interpretation in terms of processes of population growth.

scholarly journals Finiteness of Record values and Alternative Asymptotic Theory of Records with Atom Endpoints

2020 ◽  
Vol 15 (3) ◽  
pp. 2371-2385
Author(s):  
Gane Samb Lo ◽  
Harouna Sangaré ◽  
Cherif Mamadou Moctar Traoré ◽  
Mohammad Ahsanullah
Keyword(s):  
Negative Binomial ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Record Values ◽  
Cumulative Distribution ◽  
Hitting Times ◽  
Record Times ◽  
Binomial Random Variable ◽  
Upper Record Values

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for iid real valued random variable, that strong upper record values are finite if and only if the upper endpoint is finite and is an atom of the common cumulative distribution function. The only asymptotic study left to us concerns the infinite sequence of hitting times of that upper endpoints, which by the way, is the sequence of weak record times. The asymptotic characterizations are made using negative binomial random variables and the dimensional multinomial random variables. Asymptotic comparison in terms of consistency bounds and confidence intervals on the different sequences of hitting times are provided. The example of a binomial random variable is given.

On discrete time extremal processes

1974 ◽  
Vol 6 (03) ◽  
pp. 580-592 ◽  
Author(s):  
R. W. Shorrock
Keyword(s):  
Population Growth ◽  
Discrete Time ◽  
Continuous Time ◽  
Record Values ◽  
Record Times ◽  
State Spaces ◽  
Upper Record Values ◽  
Extremal Processes ◽  
Upper Record ◽  
Continuous Time Processes

Upper record values and times and inter-record times are studied in their rôles as embedded structures in discrete time extremal processes. Various continuous time approximations to the discrete-time processes are analysed, especially as processes over their state spaces. Discrete time processes, suitably normalized after crossing a threshold T, are shown to converge to limiting continuous time processes as T → ∞ under suitable assumptions on the underlying CDF F, for example, when 1 — F varies regularly at ∞, and more generally. Discrete time extremal processes viewed as processes over their state spaces are noted to have an interesting interpretation in terms of processes of population growth.

The distributions of interrecord fillings

2016 ◽  
Vol 26 (4) ◽  
Author(s):  
Oleg P. Orlov ◽  
Nikolay Yu. Pasynkov
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Record Values ◽  
Continuous Distribution Function ◽  
Record Times ◽  
Initial Sequence

AbstractIn a sequence of independent positive random variables with the same continuous distribution function a monotonic subsequence of record values is chosen. A corresponding sequence of record times divides the initial sequence into interrecord intervals. Let

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.

Records in the presence of a linear trend

1987 ◽  
Vol 19 (4) ◽  
pp. 801-828 ◽  
Author(s):  
Rocco Ballerini ◽  
Sidney I. Resnick
Keyword(s):  
Gaussian Process ◽  
Limit Theorems ◽  
Linear Trend ◽  
Random Sequence ◽  
Record Values ◽  
Stationary Gaussian Process ◽  
Record Times

Records from the sequence Yn = Xn + cn, n ≧ 1 are analyzed, where {Xn} is a strictly stationary random sequence. We prove various limit theorems for the record rate, record times, and record values. The situation when {Xn} is a stationary Gaussian process is considered with special attention given to Gaussian ARMA sequences. Data for the 400 and 800 metre races are used to illustrate these results.

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