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.

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.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

scholarly journals Contextuality in canonical systems of random variables

2017 ◽  
Vol 375 (2106) ◽  
pp. 20160389 ◽  
Author(s):  
Ehtibar N. Dzhafarov ◽  
Víctor H. Cervantes ◽  
Janne V. Kujala
Keyword(s):  
Joint Distribution ◽  
Random Variables ◽  
Canonical System ◽  
Canonical Representation ◽  
Single Pair ◽  
Canonical Systems ◽  
Joint Distributions ◽  
Content Sharing ◽  
Maximal Coupling ◽  
Quantum Revolution

Random variables representing measurements, broadly understood to include any responses to any inputs, form a system in which each of them is uniquely identified by its content (that which it measures) and its context (the conditions under which it is recorded). Two random variables are jointly distributed if and only if they share a context. In a canonical representation of a system, all random variables are binary, and every content-sharing pair of random variables has a unique maximal coupling (the joint distribution imposed on them so that they coincide with maximal possible probability). The system is contextual if these maximal couplings are incompatible with the joint distributions of the context-sharing random variables. We propose to represent any system of measurements in a canonical form and to consider the system contextual if and only if its canonical representation is contextual. As an illustration, we establish a criterion for contextuality of the canonical system consisting of all dichotomizations of a single pair of content-sharing categorical random variables. This article is part of the themed issue ‘Second quantum revolution: foundational questions’.

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.

scholarly journals A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

2008 ◽  
Vol 45 (04) ◽  
pp. 1181-1185 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Poisson Processes ◽  
Bernoulli Random Variables ◽  
Conditional Poisson

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.

On Maxima and Minima of Partial Sums of Strongly Interchangeable Random Variables

1990 ◽  
Vol 4 (3) ◽  
pp. 319-332 ◽  
Author(s):  
Teunis J. Ott ◽  
J. George Shanthikumar
Keyword(s):  
Random Vector ◽  
Queueing Networks ◽  
Joint Distribution ◽  
Random Variables ◽  
Markov Property ◽  
Partial Sums ◽  
Random Vectors ◽  
Symmetric Distributions ◽  
Voice Communication ◽  
Special Instance

We introduce the concept of “strong interchangeability” of random vectors. Strongly interchangeable random vectors arise naturally in packetized voice channels, M/G/1 queues, symmetric queueing networks, and other standard symmetric distributions. We study some properties of strongly interchangeable random vectors. We show that if (X1, …, XN) is a strongly interchangeable random vector, then even though there is no Markov property, taboo probabilities can be used to compute the joint distribution of ŽN = min1≤n≤N σnk=IXk and ZN = max1≤n≤N σnk=1Xk. For a special instance of this problem that arises in packetized voice communication, it is shown that the resulting algorithm essentially has a complexity of order N4. When ( σnk=1Xk, n = 1,… N) is an associated random vector bound for the joint distribution of ŽN and ZN are obtained and applied to the packetized voice communication problem.

scholarly journals A Note on Embedding Certain Bernoulli Sequences in Marked Poisson Processes

2008 ◽  
Vol 45 (4) ◽  
pp. 1181-1185 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Poisson Processes ◽  
Bernoulli Random Variables ◽  
Conditional Poisson

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.

Exchangeable Processes: de Finetti’s Theorem Revisited

2020 ◽  
Vol 45 (3) ◽  
pp. 1153-1163
Author(s):  
Ehud Lehrer ◽  
Dimitry Shaiderman
Keyword(s):  
Representation Theorem ◽  
Joint Distribution ◽  
Stationary Process ◽  
Infinite Sequence ◽  
Convex Combination ◽  
Random Variables ◽  
Frequency Dependent ◽  
Distributed Processes ◽  
The Relationship ◽  
Independent And Identically Distributed

A sequence of random variables is exchangeable if the joint distribution of any finite subsequence is invariant to permutations. De Finetti’s representation theorem states that every exchangeable infinite sequence is a convex combination of independent and identically distributed processes. In this paper, we explore the relationship between exchangeability and frequency-dependent posteriors. We show that any stationary process is exchangeable if and only if its posteriors depend only on the empirical frequency of past events.

Exchangeability, Representation Theorems, and Subjectivity

2011 ◽  
pp. 39-57 ◽  
Author(s):  
Dale J. Poirier
Keyword(s):  
Statistical Inference ◽  
Joint Distribution ◽  
Likelihood Function ◽  
Random Variables ◽  
Parametric Models ◽  
Bayesian Econometrics ◽  
Representation Theorems ◽  
Bernoulli Random Variables ◽  
Observable Data

This article is concerned with the foundation of statistical inference in the representation theorems. It shows how different assumptions about the joint distribution of the observable data lead to different parametric models defined by prior and likelihood function. Parametric models arise as an implication of the assumptions about observables. The article presents many extensions and offers description of the subjectivist attitude that underlies much of Bayesian econometrics. This subjectivist interpretation is close to probability. This article discusses exchangeability as the foundation for Bayesian econometrics. It serves as the basis for further extensions to incorporate heterogeneity and dependency across observations. It also discusses representation theorems involving random variables more complicated than Bernoulli random variables. They are not true properties of reality but are useful for making inferences regarding future observables.

On the locations of maxima and minima in a sequence of exchangeable random variables

2021 ◽  
Vol 105 (0) ◽  
pp. 35-50
Author(s):  
D. Ferger
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Classical Approach ◽  
Linear Transformations ◽  
Exchangeable Random Variables

We show for a finite sequence of exchangeable random variables that the locations of the maximum and minimum are independent from every symmetric event. In particular they are uniformly distributed on the grid without the diagonal. Moreover, for an infinite sequence we show that the extrema and their locations are asymptotically independent. Here, in contrast to the classical approach we do not use affine-linear transformations. Moreover it is shown how the new transformations can be used in extreme value statistics.

Sign in / Sign up

Export Citation Format

Share Document