A note on exceedances and rare events of non-stationary sequences

Exceedances of a non-stationary sequence above a boundary define certain point processes, which converge in distribution under mild mixing conditions to Poisson processes. We investigate necessary and sufficient conditions for the convergence of the point process of exceedances, the point process of upcrossings and the point process of clusters of exceedances. Smooth regularity conditions, as smooth oscillation of the non-stationary sequence, imply that these point processes converge to the same Poisson process. Since exceedances are asymptotically rare, the results are extended to triangular arrays of rare events.

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Further results for Gauss-Poisson processes

Advances in Applied Probability ◽

10.1017/s0001867800038192 ◽

1972 ◽

Vol 4 (01) ◽

pp. 151-176 ◽

Cited By ~ 13

Author(s):

R. K. Milne ◽

M. Westcott

Keyword(s):

Point Processes ◽

Sufficient Conditions ◽

Infinite Divisibility ◽

Poisson Processes ◽

Generating Functional ◽

Basic Approach ◽

New Class ◽

Two Measures ◽

Necessary And Sufficient ◽

Statistical Results

Newman (1970) introduced an interesting new class of point processes which he called Gauss-Poisson. They are characterized, in the most general case, by two measures. We determine necessary and sufficient conditions on these measures for the resulting point process to be well defined, and proceed to a systematic study of its properties. These include stationarity, ergodicity, and infinite divisibility. We mention connections with other classes of point processes and some statistical results. Our basic approach is through the probability generating functional of the process.

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Further results for Gauss-Poisson processes

Advances in Applied Probability ◽

10.2307/1425809 ◽

1972 ◽

Vol 4 (1) ◽

pp. 151-176 ◽

Cited By ~ 30

Author(s):

R. K. Milne ◽

M. Westcott

Keyword(s):

Point Processes ◽

Sufficient Conditions ◽

Infinite Divisibility ◽

Poisson Processes ◽

Generating Functional ◽

Basic Approach ◽

New Class ◽

Two Measures ◽

Necessary And Sufficient ◽

Statistical Results

Newman (1970) introduced an interesting new class of point processes which he called Gauss-Poisson. They are characterized, in the most general case, by two measures. We determine necessary and sufficient conditions on these measures for the resulting point process to be well defined, and proceed to a systematic study of its properties. These include stationarity, ergodicity, and infinite divisibility. We mention connections with other classes of point processes and some statistical results. Our basic approach is through the probability generating functional of the process.

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High-Level exceedances of regenerative and semi-stationary processes

Journal of Applied Probability ◽

10.1017/s0021900200047240 ◽

1980 ◽

Vol 17 (02) ◽

pp. 423-431 ◽

Cited By ~ 3

Author(s):

Richard Serfozo

Keyword(s):

Point Processes ◽

Stationary Process ◽

Sufficient Conditions ◽

Poisson Processes ◽

Partial Sums ◽

Stationary Processes ◽

Special Cases ◽

Cumulative Amount ◽

High Level ◽

Necessary And Sufficient

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

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High-Level exceedances of regenerative and semi-stationary processes

Journal of Applied Probability ◽

10.2307/3213031 ◽

1980 ◽

Vol 17 (2) ◽

pp. 423-431 ◽

Cited By ~ 11

Author(s):

Richard Serfozo

Keyword(s):

Point Processes ◽

Stationary Process ◽

Sufficient Conditions ◽

Poisson Processes ◽

Partial Sums ◽

Stationary Processes ◽

Special Cases ◽

Cumulative Amount ◽

High Level ◽

Necessary And Sufficient

The cumulative amount of time that a regenerative or semi-stationary process exceeds a high level and other measures of these exceedances are considered as special cases of a non-decreasing stochastic process of partial sums. We present necessary and sufficient conditions for these exceedance processes to converge in distribution to Poisson processes or processes with stationary independent non-negative increments as the level goes to infinity. We apply our results to random walks, M/M/s queues, and thinnings of point processes.

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Rarefactions of compound point processes

Journal of Applied Probability ◽

10.1017/s0021900200037426 ◽

1984 ◽

Vol 21 (04) ◽

pp. 710-719

Author(s):

Richard F. Serfozo

Keyword(s):

Poisson Process ◽

Point Process ◽

Classical Result ◽

Point Processes ◽

Rare Events ◽

Random Measure ◽

Bernoulli Trials ◽

Random Measures ◽

Infinitely Divisible ◽

Independent Increments

The Poisson process is regarded as a point process of rare events because of the classical result that the number of successes in a sequence of Bernoulli trials is asymptotically Poisson as the probability of a success tends to 0. It is shown that this rareness property of the Poisson process is characteristic of any infinitely divisible point process or random measure with independent increments. These processes and measures arise as limits of certain rarefactions of compound point processes: purely atomic random measures with uniformly null atom sizes. Examples include thinnings and partitions of point processes.

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Order statistics, Poisson processes and repairable systems

Journal of Applied Probability ◽

10.1017/s0021900200104061 ◽

1976 ◽

Vol 13 (03) ◽

pp. 519-529 ◽

Cited By ~ 1

Author(s):

Douglas R. Miller

Keyword(s):

Order Statistics ◽

Point Processes ◽

Failure Time ◽

Sufficient Conditions ◽

Renewal Processes ◽

Repairable Systems ◽

Homogeneous Poisson Process ◽

Necessary And Sufficient ◽

Non Homogeneous Poisson Process ◽

Weibull Process

Necessary and sufficient conditions are presented under which the point processes equivalent to order statistics of n i.i.d. random variables or superpositions of n i.i.d. renewal processes converge to a non-degenerate limiting process as n approaches infinity. The limiting process must be one of three types of non-homogeneous Poisson process, one of which is the Weibull process. These point processes occur as failure-time models in the reliability theory of repairable systems.

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Further results on ASTA for general stationary processes and related problems

Journal of Applied Probability ◽

10.2307/3214823 ◽

1990 ◽

Vol 27 (4) ◽

pp. 792-804 ◽

Cited By ~ 9

Author(s):

Masakiyo Miyazawa ◽

Ronald W. Wolff

Keyword(s):

Point Process ◽

Stationary Process ◽

Sufficient Conditions ◽

Jump Process ◽

Arrival Process ◽

General Process ◽

Left Hand ◽

Time Averages ◽

Semi Markov Processes ◽

Necessary And Sufficient

We consider the equivalence of state probabilities of a general stationary process at an arbitrary time and at embedded epochs of a given point process, which is called ASTA (Arrivals See Time Averages). By using an event-conditonal intensity, we give necessary and sufficient conditions for ASTA for a large class of state sets, which determines a state distribution. We do not need any additional assumptions except that the general process has left-hand limits at all points of time. Especially, for a stationary pure-jump process with a point process, ASTA is obtained for all state sets. As an application of those results, Anti-PASTA is obtained for a pure-jump Markov process and a certain class of GSMP (Generalized Semi-Markov Processes), where Anti-PASTA means that ASTA implies that the arrival process is Poisson.

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A bivariate point process connected with electronic counters

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1977.0126 ◽

1977 ◽

Vol 356 (1685) ◽

pp. 149-160 ◽

Cited By ~ 58

Keyword(s):

Point Process ◽

Point Processes ◽

Dead Time ◽

Poisson Processes ◽

Constant Time ◽

Coincidence Rate ◽

Time Period

Consider three independent Poisson processes of point events of rates λ 1 , λ 2 and λ 12 . There are two electronic counters, the first recording events from the first and third Poisson processes, and the second recording events from the second and third Poisson processes. Both counters have constant dead-time, i.e. following the recording of an event on a counter no further event can be recorded on that counter until the appropriate constant time has elapsed. Two ways of estimating λ 12 are via a coincidence rate, i.e. the rate of occurrence of pairs of events separated by less than a suitable small tolerance, and via the covariance of the numbers of events recorded on the two counters in a suitable time period. The theoretical values of these quantities are calculated allowing for dead-time. The techniques used illustrate the study of bivariate point processes.

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The Hurst effect under trends

Journal of Applied Probability ◽

10.2307/3213900 ◽

1983 ◽

Vol 20 (3) ◽

pp. 649-662 ◽

Cited By ~ 102

Author(s):

R. N. Bhattacharya ◽

Vijay K. Gupta ◽

Ed Waymire

Keyword(s):

Hurst Exponent ◽

Sufficient Conditions ◽

Random Variables ◽

Stationary Sequence ◽

Monotonic Trend ◽

Weakly Dependent Random Variables ◽

Necessary And Sufficient ◽

Hurst Effect ◽

Weakly Dependent ◽

Negative Parameter

Necessary and sufficient conditions for the so-called Hurst effect are given in the case of a weakly dependent stationary sequence of random variables perturbed by a trend. As a consequence of this general result it is shown that the Hurst effect is present in the case of weakly dependent random variables with a small monotonic trend of the form f(n) = c(m + n)ß, where m is an arbitrary non-negative parameter and c is not 0. For – ½ < ß < 0 the Hurst exponent is shown to be precisely given by 1 + ß. For ß ≦ – ½ and for ß = 0 the Hurst exponent is 0.5, while for ß > 0 it is 1. This simple mathematical model, motivated by empirical evidence in various geophysical records, demonstrates the presence of the Hurst effect in a direction not explored before.

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