Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses

1971 ◽  
Vol 8 (1) ◽  
pp. 170-183 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Poisson Process ◽  
Stationary Point ◽  
Joint Distribution ◽  
Initial Conditions ◽  
Response Process ◽  
New Approach ◽  
Dependency Structure ◽  
Joint Distributions ◽  
Selective Interaction ◽  
Stationary Point Process

This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.

Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses

1971 ◽  
Vol 8 (01) ◽  
pp. 170-183 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Poisson Process ◽  
Stationary Point ◽  
Joint Distribution ◽  
Initial Conditions ◽  
Response Process ◽  
New Approach ◽  
Dependency Structure ◽  
Joint Distributions ◽  
Selective Interaction ◽  
Stationary Point Process

This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.

Selective interaction of a Poisson and renewal process: first-order stationary point results

1970 ◽  
Vol 7 (02) ◽  
pp. 359-372 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Stationary Point ◽  
Joint Distribution ◽  
Point Processes ◽  
First Order ◽  
Response Interval ◽  
Selective Interaction ◽  
Counting Distribution ◽  
Stationary Point Process ◽  
Equilibrium Process ◽  
Stationary Point Processes

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

Selective interaction of a Poisson and renewal process: the spectrum of the intervals between responses

1971 ◽  
Vol 8 (04) ◽  
pp. 731-744
Author(s):  
A. J. Lawrance
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Stationary Point ◽  
Point Theory ◽  
Time Function ◽  
Qualitative Behavior ◽  
Average Response ◽  
Joint Distributions ◽  
Selective Interaction ◽  
Serial Correlations

This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.

Selective interaction of a Poisson and renewal process: first-order stationary point results

1970 ◽  
Vol 7 (2) ◽  
pp. 359-372 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Stationary Point ◽  
Joint Distribution ◽  
Point Processes ◽  
First Order ◽  
Response Interval ◽  
Selective Interaction ◽  
Counting Distribution ◽  
Stationary Point Process ◽  
Equilibrium Process ◽  
Stationary Point Processes

The simple stationarity of a previously derived equilibrium process of responses in a renewal inhibited stationary point process is established by deriving the joint distribution of the number of responses in contiguous intervals in the process. For a renewal inhibited Poisson process the variancetime function of the process is obtained; the distribution of an arbitrary between-response interval and the synchronous counting distribution are also derived following analytic justification of the required results. These results strengthen earlier results in the theory of stationary point processes. Three other point processes arising from the interaction are briefly discussed.

Selective interaction of a Poisson and renewal process: the spectrum of the intervals between responses

1971 ◽  
Vol 8 (4) ◽  
pp. 731-744
Author(s):  
A. J. Lawrance
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Stationary Point ◽  
Point Theory ◽  
Time Function ◽  
Qualitative Behavior ◽  
Average Response ◽  
Joint Distributions ◽  
Selective Interaction ◽  
Serial Correlations

This paper is concerned with the spectrum of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b), (1971)). A generating function for all the pairwise joint distributions of the synchronous intervals following an average response is obtained and leads directly to the associated serial correlations. It is shown that these correlations are equivalent to those predicted on different assumptions by the general stationary point theory. The results are then used to obtain the interval spectrum, and to exhibit a relationship between the sum of the serial correlations and the variance-time function. Explicit results for the spectrum of the renewal inhibited Poisson process are given for gamma inhibitory distributions, and the qualitative behavior is determined. Possible further developments are briefly discussed.

scholarly journals Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

2012 ◽  
Vol 49 (3) ◽  
pp. 758-772 ◽  
Author(s):  
Fred W. Huffer ◽  
Jayaram Sethuraman
Keyword(s):  
Poisson Process ◽  
Random Vector ◽  
Joint Distribution ◽  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Factorial Moments ◽  
Bernoulli Random Variables ◽  
Joint Distributions ◽  
Bernoulli Sequence

An infinite sequence (Y1, Y2,…) of independent Bernoulli random variables with P(Yi = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z1, Z2, Z3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z1, Z2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y1, Y2,…, Yn) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

scholarly journals Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

2012 ◽  
Vol 49 (03) ◽  
pp. 758-772 ◽  
Author(s):  
Fred W. Huffer ◽  
Jayaram Sethuraman
Keyword(s):  
Poisson Process ◽  
Random Vector ◽  
Joint Distribution ◽  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Factorial Moments ◽  
Bernoulli Random Variables ◽  
Joint Distributions ◽  
Bernoulli Sequence

An infinite sequence (Y 1, Y 2,…) of independent Bernoulli random variables with P(Y i = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z 1, Z 2, Z 3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z 1, Z 2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y 1, Y 2,…, Y n ) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z 1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

Properties of the bivariate delayed Poisson process

1975 ◽  
Vol 12 (02) ◽  
pp. 257-268 ◽  
Author(s):  
A. J. Lawrance ◽  
P. A. W. Lewis
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Initial Conditions ◽  
Arbitrary Time ◽  
Delay Distribution ◽  
Counting Distribution

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

Selective interaction of a stationary point process and a renewal process

1970 ◽  
Vol 7 (02) ◽  
pp. 483-489 ◽  
Author(s):  
A. J. Lawrance
Keyword(s):  
Explicit Expression ◽  
Point Process ◽  
Renewal Process ◽  
Stationary Point ◽  
Equilibrium Conditions ◽  
Selective Interaction ◽  
Counting Distribution ◽  
Stationary Point Process

The selective interaction of a stationary point process and a renewal process is studied. Equilibrium conditions for the resulting point process are given, the equilibrium counting distribution is derived, and an explicit expression for the rate of the process is determined.

Properties of the bivariate delayed Poisson process

1975 ◽  
Vol 12 (2) ◽  
pp. 257-268 ◽  
Author(s):  
A. J. Lawrance ◽  
P. A. W. Lewis
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Joint Distribution ◽  
Poisson Point Process ◽  
Initial Conditions ◽  
Arbitrary Time ◽  
Delay Distribution ◽  
Counting Distribution

The bivariate Poisson point process introduced in Cox and Lewis (1972), and there called the bivariate delayed Poisson process, is studied further; the process arises from pairs of delays on the events of a Poisson process. In particular, results are obtained for the stationary initial conditions, the joint distribution of the number of operative delays at an arbitrary time, the asynchronous counting distribution, and two semi-synchronous interval distributions. The joint delay distribution employed allows for dependence and two-sided delays, but the model retains the independence between different pairs of delays.

Sign in / Sign up

Export Citation Format

Share Document