### Abstract

The definition of pseudo-Poissonian processes is given in the famous monograph of William Feller (1971, Vol. II, Chapter X). The contemporary development of the theory of information flows generates new interest in the detailed analysis of behavior and characteristics of pseudo-Poissonian processes. Formally, a pseudo-Poissonian process is a Poissonian subordination of the mathematical time of an independent random sequence (the time randomization of a random sequence). We consider a sequence consisting of independent identically distributed random variables with second moments. In this case, pseudo-Poissonian processes do not have independent increments, but it is possible to calculate the autocovariance function, and it turns out that it exponentially decreases. Appropriately normed sums of independent copies of such pseudo-Poissonian processes tend to the Ornstein–Uhlenbeck process. A generalization of driving Poissonian processes to the case where the intensity is random is considered and it is shown that, under this generalization, the autocovariance function of the corresponding pseudo-Poissonian process is the Laplace transform of the distribution of that random intensity. Stochastic choice principles for the distribution of the random intensity are shortly discussed and they are illustrated by two detailed examples.

Original language | English |
---|---|

Pages (from-to) | 153-160 |

Number of pages | 8 |

Journal | Vestnik St. Petersburg University: Mathematics |

Volume | 50 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2017 |

### Keywords

- Laplace transform of distributions
- processes of the Ornstein–Uhlenbeck type
- pseudo-Poissonian processes
- random intensity
- stability

### Scopus subject areas

- Mathematics(all)

### Cite this

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**Pseudo-Poissonian processes with stochastic intensity and a class of processes generalizing the Ornstein–Uhlenbeck process.** / Rusakov, O. V.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Pseudo-Poissonian processes with stochastic intensity and a class of processes generalizing the Ornstein–Uhlenbeck process

AU - Rusakov, O. V.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - The definition of pseudo-Poissonian processes is given in the famous monograph of William Feller (1971, Vol. II, Chapter X). The contemporary development of the theory of information flows generates new interest in the detailed analysis of behavior and characteristics of pseudo-Poissonian processes. Formally, a pseudo-Poissonian process is a Poissonian subordination of the mathematical time of an independent random sequence (the time randomization of a random sequence). We consider a sequence consisting of independent identically distributed random variables with second moments. In this case, pseudo-Poissonian processes do not have independent increments, but it is possible to calculate the autocovariance function, and it turns out that it exponentially decreases. Appropriately normed sums of independent copies of such pseudo-Poissonian processes tend to the Ornstein–Uhlenbeck process. A generalization of driving Poissonian processes to the case where the intensity is random is considered and it is shown that, under this generalization, the autocovariance function of the corresponding pseudo-Poissonian process is the Laplace transform of the distribution of that random intensity. Stochastic choice principles for the distribution of the random intensity are shortly discussed and they are illustrated by two detailed examples.

AB - The definition of pseudo-Poissonian processes is given in the famous monograph of William Feller (1971, Vol. II, Chapter X). The contemporary development of the theory of information flows generates new interest in the detailed analysis of behavior and characteristics of pseudo-Poissonian processes. Formally, a pseudo-Poissonian process is a Poissonian subordination of the mathematical time of an independent random sequence (the time randomization of a random sequence). We consider a sequence consisting of independent identically distributed random variables with second moments. In this case, pseudo-Poissonian processes do not have independent increments, but it is possible to calculate the autocovariance function, and it turns out that it exponentially decreases. Appropriately normed sums of independent copies of such pseudo-Poissonian processes tend to the Ornstein–Uhlenbeck process. A generalization of driving Poissonian processes to the case where the intensity is random is considered and it is shown that, under this generalization, the autocovariance function of the corresponding pseudo-Poissonian process is the Laplace transform of the distribution of that random intensity. Stochastic choice principles for the distribution of the random intensity are shortly discussed and they are illustrated by two detailed examples.

KW - Laplace transform of distributions

KW - processes of the Ornstein–Uhlenbeck type

KW - pseudo-Poissonian processes

KW - random intensity

KW - stability

UR - http://www.scopus.com/inward/record.url?scp=85022069682&partnerID=8YFLogxK

U2 - 10.3103/S106345411702011X

DO - 10.3103/S106345411702011X

M3 - Article

VL - 50

SP - 153

EP - 160

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 2

ER -