A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

Holst, Lars

doi:10.1017/s0021900200005052

Search citation statements

Order By: Relevance

Paper Sections

Select...

Counts Of Non-records Between Records1

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2009

2012

Publication Types

Select...

Article3

Relationship

Self Cite2

Independent1

Authors

Journals

Cited by 3 publications

(2 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To our knowledge the embedding approach used above is new. More general cases using embedding are considered in [9] and [11].…”

Section: Counts Of Non-records Between Recordsmentioning

confidence: 99%

A note on records in a random sequence

Holst

2011

Ark. Mat.

Self Cite

View full text Add to dashboard Cite

show abstract

“…To our knowledge the embedding approach used above is new. More general cases using embedding are considered in [9] and [11].…”

Section: Counts Of Non-records Between Recordsmentioning

confidence: 99%

A note on records in a random sequence

Holst

2011

Ark. Mat.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Holst (2007) extended this result to Bern(a, b) by using a Pólya-Hoppe urn argument. Holst (2008a), Huffer et al (2008) used beta random variables to construct a CMPP leading to the Bern(a, b) sequence.…”

Section: Introductionmentioning

confidence: 99%

Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

Huffer

Sethuraman

2012

J. Appl. Probab.

View full text Add to dashboard Cite

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.

show abstract

On Consecutive Records in Certain Bernoulli Sequences

Holst

2009

Journal of Applied Probability

Self Cite

View full text Add to dashboard Cite

In an infinite sequence of independent Bernoulli trials with success probabilities pk=a/(a+b +k-1) for k=1,2,3,…, let Nr be the number of r≥2 consecutive successes. Expressions for the first two moments of Nr are derived. Asymptotics of the probability of no occurrence of r consecutive successes for large r are obtained. Using an embedding in a marked Poisson process, it is indicated how the distribution of Nr can be calculated for small r.

show abstract

A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

Abstract: A sequence of independent Bernoulli random variables with success probabilities a/(a + b + k − 1), k = 1, 2, 3, . . . , is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

Cited by 3 publications

References 5 publications

A note on records in a random sequence

A note on records in a random sequence

Joint Distributions of Counts of Strings in Finite Bernoulli Sequences

On Consecutive Records in Certain Bernoulli Sequences

Contact Info

Product

Resources

About