On the distribution of the number of cyclically occurring random events

1972 ◽  
Vol 9 (03) ◽  
pp. 681-683
Author(s):  
Leon Podkaminer
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Time Intervals ◽  
Time Period ◽  
Random Events ◽  
Mutually Independent

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.

On the distribution of the number of cyclically occurring random events

1972 ◽  
Vol 9 (3) ◽  
pp. 681-683
Author(s):  
Leon Podkaminer
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Time Intervals ◽  
Time Period ◽  
Random Events ◽  
Mutually Independent

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (2) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (02) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T 1, T 1 + T 2, … it undergoes jumps ξ 1, ξ 2, …, where the time intervals T 1, T 2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi , are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

Ehrenfest urn models

1965 ◽  
Vol 2 (02) ◽  
pp. 352-376 ◽  
Author(s):  
Samuel Karlin ◽  
James McGregor
Keyword(s):  
Exponential Distribution ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Urn Models ◽  
Time Intervals ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Random Times

In the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in stateiif there areiballs in urn I, N −iballs in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distributionWhen an event occurs a ball is chosen at random (each of theNballs has probability 1/Nto be chosen), removed from its urn, and then placed in urn I with probabilityp, in urn II with probabilityq= 1 −p, (0 <p< 1).

Markov chains governed by complicated renewal processes

1970 ◽  
Vol 2 (02) ◽  
pp. 287-322
Author(s):  
T. Gergely ◽  
I. N. Tsukanow ◽  
I. I. Yezhow
Keyword(s):  
Markov Chains ◽  
Sojourn Time ◽  
Practical Importance ◽  
Service Systems ◽  
Independent Random Variables ◽  
Renewal Processes ◽  
Time Intervals ◽  
Mass Service ◽  
Fixed Region ◽  
Mutually Independent

In this work Markov chains governed by complicated processes are introduced and investigated (Section 1). In Section 2 an ergodic theorem for these processes is formulated, while in Section 3 the sojourn time of the process in a fixed region is studied; in Section 4 some examples are considered. The processes studied are of practical importance in the description of mass service systems and the theory of reliability for which the time intervals between successive demands cannot be assumed to be mutually independent random variables. It is shown that the dependence parameter r of these processes, if it is sufficiently large, allows us to formulate a relationship between the time intervals in question.

A joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (01) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X 1, X 2, … are non-constant mutually independent random variables with X 1,X 2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X 1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

A joint characterization of the multinomial distribution and the Poisson process

1983 ◽  
Vol 20 (1) ◽  
pp. 202-208 ◽  
Author(s):  
George Kimeldorf ◽  
Peter F. Thall
Keyword(s):  
Poisson Process ◽  
Present Article ◽  
Point Process ◽  
Random Variables ◽  
Multinomial Distribution ◽  
Process Theory ◽  
Independent Random Variables ◽  
Mutually Independent

It has been recently proved that if N, X1, X2, … are non-constant mutually independent random variables with X1,X2, … identically distributed and N non-negative and integer-valued, then the independence of and implies that X1 is Bernoulli and N is Poisson. A well-known theorem in point process theory due to Fichtner characterizes a Poisson process in terms of a sum of independent thinnings. In the present article, simultaneous generalizations of both of these results are provided, including a joint characterization of the multinomial distribution and the Poisson process.

On a Problem of Fluctuations of Sums of Independent Random Variables

1975 ◽  
Vol 12 (S1) ◽  
pp. 29-37
Author(s):  
Lajos Takács
Keyword(s):  
Limit Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Discrete Random Variables ◽  
Mutually Independent ◽  
Independent And Identically Distributed

The author determines the distribution and the limit distribution of the number of partial sums greater than k (k = 0, 1, 2, …) for n mutually independent and identically distributed discrete random variables taking on the integers 1, 0, − 1, − 2, ….

Markov chains governed by complicated renewal processes

1970 ◽  
Vol 2 (2) ◽  
pp. 287-322 ◽  
Author(s):  
T. Gergely ◽  
I. N. Tsukanow ◽  
I. I. Yezhow
Keyword(s):  
Markov Chains ◽  
Sojourn Time ◽  
Practical Importance ◽  
Service Systems ◽  
Independent Random Variables ◽  
Renewal Processes ◽  
Time Intervals ◽  
Mass Service ◽  
Fixed Region ◽  
Mutually Independent

In this work Markov chains governed by complicated processes are introduced and investigated (Section 1). In Section 2 an ergodic theorem for these processes is formulated, while in Section 3 the sojourn time of the process in a fixed region is studied; in Section 4 some examples are considered. The processes studied are of practical importance in the description of mass service systems and the theory of reliability for which the time intervals between successive demands cannot be assumed to be mutually independent random variables. It is shown that the dependence parameter r of these processes, if it is sufficiently large, allows us to formulate a relationship between the time intervals in question.

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