# How many digits are needed?

Methodology and Computing in Applied Probability

Published On 2024/3

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

Journal

Methodology and Computing in Applied Probability

Published On

2024/3

Volume

26

Issue

1

Page

5

## Authors

#### Jesper Møller

##### Aalborg Universitet

Position

Professor in Statistics

H-Index(all)

46

H-Index(since 2020)

23

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Statistics

Probability Theory

University Profile Page

#### Ira Herbst

##### University of Virginia

Position

Professor of Mathematics

H-Index(all)

36

H-Index(since 2020)

14

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Physics

University Profile Page

### Other Articles from authors

Ira Herbst

University of Virginia

arXiv preprint arXiv:2404.08387

##### The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let be the base- expansion of a continuous random variable on the unit interval where is the positive solution to for an integer (i.e., is a generalization of the golden mean for which ). We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2024/4/12*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.09525

##### Coupling results and Markovian structures for number representations of continuous random variables

A general setting for nested subdivisions of a bounded real set into intervals defining the digits of a random variable with a probability density function is considered. Under the weak condition that is almost everywhere lower semi-continuous, a coupling between and a non-negative integer-valued random variable is established so that have an interpretation as the ``sufficient digits'', since the distribution of conditioned on does not depend on . Adding a condition about a Markovian structure of the lengths of the intervals in the nested subdivisions, becomes a Markov chain of a certain order . If then are IID with a known distribution. When and the Markov chain is uniformly geometric ergodic, a coupling is established between and a random time so that the chain after time is stationary and follows a simple known distribution. The results are related to several examples of number representations generated by a dynamical system, including base- expansions, generalized L\"uroth series, -expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.

*2024/4/15*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.08387

##### The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let be the base- expansion of a continuous random variable on the unit interval where is the positive solution to for an integer (i.e., is a generalization of the golden mean for which ). We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2024/4/12*

Ira Herbst

University of Virginia

arXiv preprint arXiv:2312.09652

##### The asymptotic distribution of the remainder in a certain base- expansion

Let be the base- expansion of a continuous random variable on the unit interval where is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2023/12/15*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2312.09652

##### The asymptotic distribution of the remainder in a certain base- expansion

Let be the base- expansion of a continuous random variable on the unit interval where is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

*2023/12/15*

Jesper Møller

Aalborg Universitet

Proceedings of the London Mathematical Society

##### Realizability and tameness of fusion systems

A saturated fusion system over a finite p$p$‐group S$S$ is a category whose objects are the subgroups of S$S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over S$S$ is realized by a finite group G$G$ if S$S$ is a Sylow p$p$‐subgroup of G$G$ and morphisms in the category are those induced by conjugation in G$G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the …

*2023/12*

Jesper Møller

Aalborg Universitet

ACM Transactions on Spatial Algorithms and Systems

##### Stochastic Routing with Arrival Windows

Arriving at a destination within a specific time window is important in many transportation settings. For example, trucks may be penalized for early or late arrivals at compact terminals, and early and late arrivals at general practitioners, dentists, and so on, are also discouraged, in part due to COVID. We propose foundations for routing with arrival-window constraints. In a setting where the travel time of a road segment is modeled by a probability distribution, we define two problems where the aim is to find a route from a source to a destination that optimizes or yields a high probability of arriving within a time window while departing as late as possible. In this setting, a core challenge is to enable comparison between paths that may potentially be part of a result path with the goal of determining whether a path is uninteresting and can be disregarded given the existence of another path. We show that existing solutions …

*2023/11/21*

Jesper Møller

Aalborg Universitet

Spatial Statistics

##### Fitting the grain orientation distribution of a polycrystalline material conditioned on a Laguerre tessellation

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel …

*2023/6/1*

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

##### Singular distribution functions for random variables with stationary digits

Let F be the cumulative distribution function (CDF) of the base-q expansion , where is an integer and is a stationary stochastic process with state space . In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

*2023/3*

Ira Herbst

University of Virginia

arXiv preprint arXiv:2305.16745

##### The Howland-Kato Commutator Problem II

We continue the search, begun by Kato, for all pairs of real, bounded, measurable functions that result in a positive commutator . We prove a number of partial results including a connection with Loewner's celebrated theorem on matrix monotone functions.

*2023/5/26*

Ira Herbst

University of Virginia

Methodology and Computing in Applied Probability

##### Singular distribution functions for random variables with stationary digits

Let F be the cumulative distribution function (CDF) of the base-q expansion , where is an integer and is a stationary stochastic process with state space . In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

*2023/3*

Ira Herbst

University of Virginia

Journal of Applied Probability

##### Characterization of random variables with stationary digits

Let be an integer, a stochastic process with state space , and F the cumulative distribution function (CDF) of . We show that stationarity of is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

*2022/12*

Ira Herbst

University of Virginia

arXiv preprint arXiv:2209.11850

##### Griffiths inequalities for non-interacting rotors

An old open problem is to prove correlation inequalities Efg≥(Ef)(Eg) where the expectation is in the ferromagnetic measure

*2022/9/23*

Ira Herbst

University of Virginia

arXiv preprint arXiv:2209.06189

##### Weak solutions to the Navier-Stokes equations

We prove the equivalence of being a Leray-Hopf weak solution to the Navier-Stokes equations in , to satisfying a well known integral equation. We use this equation to derive some properties of these weak solutions.

*2022/9/13*

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2212.08402

##### Cox processes driven by transformed Gaussian processes on linear networks

There is a lack of point process models on linear networks. For an arbitrary linear network, we use isotropic covariance functions with respect to the geodesic metric or the resistance metric to construct new models for isotropic Gaussian processes and hence new models for various Cox processes with isotropic pair correlation functions. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.

*2022/12/16*

Jesper Møller

Aalborg Universitet

Stat

##### Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.

*2022/12*

Jesper Møller

Aalborg Universitet

International Statistical Review

##### Should we condition on the number of points when modelling spatial point patterns?

We discuss the practice of directly or indirectly assuming a model for the number of points when modelling spatial point patterns even though it is rarely possible to validate such a model in practice because most point pattern data consist of only one pattern. We therefore explore the possibility to condition on the number of points instead when fitting and validating spatial point process models. In a simulation study with different popular spatial point process models, we consider model validation using global envelope tests based on functional summary statistics. We find that conditioning on the number of points will for some functional summary statistics lead to more narrow envelopes and thus stronger tests and that it can also be useful for correcting for some conservativeness in the tests when testing composite hypothesis. However, for other functional summary statistics, it makes little or no difference to condition …

*2022/12*

Jesper Møller

Aalborg Universitet

Journal of Applied Probability

##### Characterization of random variables with stationary digits

Let be an integer, a stochastic process with state space , and F the cumulative distribution function (CDF) of . We show that stationarity of is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

*2022/12*

### Other articles from Methodology and Computing in Applied Probability journal

Krzysztof Dębicki

Uniwersytet Wroclawski

Methodology and Computing in Applied Probability

##### On Berman Functions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}$$\end{document} with a standard fractional Brownian motion (fBm) with Hurst parameter and define for x non-negative the Berman function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}}{ \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{aligned …

*2024/3*

Gracia Yunruo Dong

University of Waterloo

Methodology and Computing in Applied Probability

##### Randomized quasi-Monte Carlo methods on triangles: extensible lattices and sequences

Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of …

*2024/6*

Christiane Lemieux

University of Waterloo

Methodology and Computing in Applied Probability

##### Randomized quasi-Monte Carlo methods on triangles: extensible lattices and sequences

Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of …

*2024/6*

Farzad Eskandari

Allameh Tabataba'i University

Methodology and Computing in Applied Probability

##### The Multivariate Generalized Linear Hawkes Process in High Dimensions with Applications in Neuroscience

The Hawkes process models have been recently become a popular tool for modeling and analysis of neural spike trains. In this article, motivated by neuronal spike trains study, we propose a novel multivariate generalized linear Hawkes process model, where covariates are included in the intensity function. We consider the problem of simultaneous variable selection and estimation for the multivariate generalized linear Hawkes process in the high-dimensional regime. Estimation of the intensity function of the high-dimensional point process is considered within a nonparametric framework, applying B-splines and the SCAD penalty for matters of sparsity. We apply the Doob-Kolmogorov inequality and the martingale central limit theory to establish the consistency and asymptotic normality of the resulting estimators. Finally, we illustrate the performance of our proposal through simulation and demonstrate its utility by …

*2024/3*

Marius Hofert

University of Waterloo

Methodology and Computing in Applied Probability

##### Randomized quasi-Monte Carlo methods on triangles: extensible lattices and sequences

Two constructions were recently proposed for constructing low-discrepancy point sets on triangles. One is based on a finite lattice, the other is a triangular van der Corput sequence. We give a continuation and improvement of these methods. We first provide an extensible lattice construction for points in the triangle that can be randomized using a simple shift. We then examine the one-dimensional projections of the deterministic triangular van der Corput sequence and quantify their sub-optimality compared to the lattice construction. Rather than using scrambling to address this issue, we show how to use the triangular van der Corput sequence to construct a stratified sampling scheme. We show how stratified sampling can be used as a more efficient implementation of nested scrambling, and that nested scrambling is a way to implement an extensible stratified sampling estimator. We also provide a test suite of …

*2024/6*

Zbigniew Michna

Uniwersytet Ekonomiczny we Wroclawiu

Methodology and Computing in Applied Probability

##### On Berman Functions

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z(t)= \exp \left( \sqrt{ 2} B_H(t)- \left|t \right|^{2H}\right) , t\in \mathbb {R}$$\end{document} with a standard fractional Brownian motion (fBm) with Hurst parameter and define for x non-negative the Berman function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {B}_{Z}(x)= \mathbb {E} \left\{ \frac{ \mathbb {I} \{ \epsilon _0(RZ) > x\}}{ \epsilon _0(RZ)}\right\} \in (0,\infty ), \end{aligned …

*2024/3*

Ira Herbst

University of Virginia

Methodology and Computing in Applied Probability

##### How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

*2024/3*

Nour Eddine BERRAHOU

Université Cadi Ayyad

Methodology and Computing in Applied Probability

##### The Bahadur Representation for Empirical and Smooth Quantile Estimators Under Association

In this paper, the Bahadur representation of the empirical and Bernstein polynomials estimators of the quantile function based on associated sequences are investigated. The rate of approximation depends on the rate of decay in covariances, and it converges to the optimal rate observed under independence when the covariances quickly approach zero. As an application, we establish a Berry-Esseen bound with the rate assuming polynomial decay of covariances. All these results are established under fairly general conditions on the underlying distributions. Additionally, we perform Monte Carlo simulations to evaluate the finite sample performance of the suggested estimators, utilizing an associated and non-mixing model.

*2024/6*

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

##### How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

*2024/3*

Maxim Finkelstein

University of the Free State

Methodology and Computing in Applied Probability

##### On survival of coherent systems subject to random shocks

We consider coherent systems subject to random shocks that can damage a random number of components of a system. Based on the distribution of the number of failed components, we discuss three models, namely, (i) a shock can damage any number of components (including zero) with the same probability, (ii) each shock damages, at least, one component, and (iii) a shock can damage, at most, one component. Shocks arrival times are modeled using three important counting processes, namely, the Poisson generalized gamma process, the Poisson phase-type process and the renewal process with matrix Mittag-Leffler distributed inter-arrival times. For the defined shock models, we discuss relevant reliability properties of coherent systems. An optimal replacement policy for repairable systems is considered as an application of the proposed modeling.

*2024/3*

Nil Kamal Hazra

Indian Institute of Technology Jodhpur

Methodology and Computing in Applied Probability

##### On survival of coherent systems subject to random shocks

We consider coherent systems subject to random shocks that can damage a random number of components of a system. Based on the distribution of the number of failed components, we discuss three models, namely, (i) a shock can damage any number of components (including zero) with the same probability, (ii) each shock damages, at least, one component, and (iii) a shock can damage, at most, one component. Shocks arrival times are modeled using three important counting processes, namely, the Poisson generalized gamma process, the Poisson phase-type process and the renewal process with matrix Mittag-Leffler distributed inter-arrival times. For the defined shock models, we discuss relevant reliability properties of coherent systems. An optimal replacement policy for repairable systems is considered as an application of the proposed modeling.

*2024/3*

Savas Dayanik

Bilkent Üniversitesi

Methodology and Computing in Applied Probability

##### Model misspecification in discrete time Bayesian online change detection

We revisit the classical formulation of the discrete time Bayesian online change detection problem in which the common distribution of an observed sequence of random variables changes at an unknown point in time. The objective is to detect the change with a stopping time of the observations and minimize a given Bayes risk. When the change time has a zero-modified geometric prior distribution, the first crossing time of the odds-ratio process over a threshold is known to be an optimal solution. In the current paper, we consider a modeler who misspecifies some of the elements of this formulation. Because of this misspecification, the modeler computes a wrong stopping threshold and follows an incorrect odds-ratio process in implementation. To find her actual Bayes risk, which is different from the value function evaluated with the wrong choices, one needs to compute the expected costs accumulated by the true …

*2023/3*

Sonya Javadi

Dogus Üniversitesi

Methodology and Computing in Applied Probability

##### On computing the multivariate poisson probability distribution

Within the theory of non-negative integer valued multivariate infinitely divisible distributions, the multivariate Poisson distribution plays a key role. As in the univariate case, any non-negative integer valued infinitely divisible multivariate distribution can be approximated by a multivariate distribution belonging to the compound Poisson family. The multivariate Poisson distribution is an important member of this family. In recent years, the multivariate Poisson distributions also has gained practical importance, since they serve as models to describe counting data having a positive covariance structure. However, due to the computational complexity of computing the multivariate Poisson probability mass function (pmf) and its corresponding cumulative distribution function (cdf), their use within these counting models is limited. Since most of the theoretical properties of the multivariate Poisson probability distribution seem …

*2023/9*

Pablo Azcue

Universidad Torcuato di Tella

Methodology and Computing in Applied Probability

##### Optimal Strategies in a Production Inventory Control Model

We consider a production-inventory control model with finite capacity and two different production rates, assuming that the cumulative process of customer demand is given by a compound Poisson process. It is possible at any time to switch over from the different production rates but it is mandatory to switch-off when the inventory process reaches the storage maximum capacity. We consider holding, production, shortage penalty and switching costs. This model was introduced by Doshi, Van Der Duyn Schouten and Talman in 1978. In their paper they found a formula for the long-run average expected cost per unit time as a function of two critical levels, in this paper we consider expected discounted cumulative costs instead. We seek to minimize this discounted cost over all admissible switching strategies. We show that the optimal cost functions for the different production rates satisfy the corresponding Hamilton …

*2023/3*

Sidney Resnick

Cornell University

Methodology and Computing in Applied Probability

##### Poisson edge growth and preferential attachment networks

When modeling a directed social network, one choice is to use the traditional preferential attachment model, which generates power-law tail distributions. In traditional directed preferential attachment, every new edge is added sequentially into the network. However, real datasets often have only coarse timestamps, which means several new edges are created at the same timestamp. Previous analyses on the evolution of social networks reveal that after reaching a stable phase, the growth of edge counts in a network follows a non-homogeneous Poisson process with a constant rate across the day but varying rates from day to day. Taking such empirical observations into account, we propose a modified preferential attachment model with Poisson edge growth, and study its asymptotic behavior. This new model is then fitted to real datasets using an extreme value estimation approach.

*2023/3*

Xuejun Wang

Anhui University

Methodology and Computing in Applied Probability

##### Strong Convergence for Weighted Sums of Widely Orthant Dependent Random Variables and Applications

In this paper, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of widely orthant dependent random variables are presented. Some applications to simple linear errors-in-variables model, nonparametric regression model, and quasi-renewal counting process are provided. Simulation studies are also carried out to confirm the theoretical results.

*2023/3*

Spiros Dafnis

Agricultural University of Athens

Methodology and Computing in Applied Probability

##### Distributions Related to Weak Runs With a Minimum and a Maximum Number of Successes: A Unified Approach

In the present paper we extend the definition of r-weak runs in sequences of binary trials so that such a run contains both a minimum and a maximum number of successes and we study the distribution of the statistic enumerating such (r-weak) runs. We are also investigating the distribution of the total number of successes in all the (r-weak) runs that are enumerated. The new introduced distributions may be of great applicability in scientific fields such as Agriculture. Our study requires an appropriate generalization of the Markov chain imbedding technique. More specifically, we introduce and study the family of Markov chain imbeddable variables of returnable-polynomial type, which generalizes the families of Markov chain imbeddable variables of binomial type, returnable type and polynomial type. The new family allows a unified approach for the study of the distributions of the statistics of current interest and other …

*2023/3*

Gabriel Martos Venturini

Universidad Torcuato di Tella

Methodology and Computing in Applied Probability

##### Level Sets Semimetrics for Probability Measures with applications in hypothesis testing

In this paper we introduce a novel family of level sets semimetrics for density functions and address subtleties entailed in the estimation and computation of such semimetrics. Given data drawn from f and q, two unknown density functions, we consider different level set semimetrics so to test the null hypothesis . The performance of such testing procedure is showcased in a Monte Carlo simulation study. Using the methods developed in the paper, we assess differences in gene expression profiles between two groups of patients with different respiratory recovery patterns in a clinical study; and find significant differences between the 15 top–ranked genes density profiles corresponding to the two groups.

*2023/3*

Sujit Kumar Samanta

National Institute of Technology, Raipur

Methodology and Computing in Applied Probability

##### Detailed Analytical and Computational Studies of D-BMAP/D-BMSP/1 Queueing System

This paper studies a discrete-time single server batch arrival and batch service queueing model with unlimited waiting space. The discrete-time batch Markovian arrival process and discrete-time batch Markovian service process, respectively, manage the arrival and service processes. We adopt the UL-type RG-factorization approach based on censoring methodology for variable size batch service queue to calculate the stationary probability vectors of the transition probability matrix with general structure Markov chain at outside observer’s epoch. We reblock the transition probability matrix to its desired M/G/1 structure to find the stationary probability vectors at outside observer’s epoch for fixed size batch service queue using the matrix analytic method. We also develop relationships to determine probability vector expressions for other important time epochs such as pre-arrival, intermediate, post-departure, and …

*2023/3*

N Balakrishnan

McMaster University

Methodology and Computing in Applied Probability

##### Joint Reliability of Two Consecutive-(1, l) or (2, k)-out-of-(2, n): F Type Systems and Its Application in Smart Street Light Deployment

To investigate some complex practical systems, such as a smart street light system composed of symmetrically deployed lighting and sensing equipment on both sides of a road segment, two consecutive-(1,l) or (2,k)-out-of-(2,n): F type systems that share components are quite useful and are therefore studied in this paper. The joint reliability of the two systems is presented by using the finite Markov chain imbedding approach (FMCIA), which also presents a new computational method for joint signatures of such systems. Compared to the direct method, the new method is not only computationally more efficient, but also presents a unified mathematical form. Finally, some numerical examples are presented to show the computational process, and some further applications and extensions of the models and the methods developed here are mentioned.

*2023/3*