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

arXiv preprint arXiv:2404.09525

Published On 2024/4/15

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.

Journal

arXiv preprint arXiv:2404.09525

Published On

2024/4/15

## Authors

#### Jesper Møller

##### Aalborg Universitet

Position

Professor in Statistics

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Mathematical Statistics

Probability Theory

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### Other Articles from authors

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*

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*

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*

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

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*

Jesper Møller

Aalborg Universitet

Spatial Statistics

##### Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points, we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points, we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step, parameter estimation based on maximum pseudolikelihood methods is tractable. For model selection, we consider maximized log pseudolikelihood functions for models of the radii conditioned on the points. Model checking is performed using global envelopes and corresponding tests in both steps and moreover by comparing observed and simulated tessellation characteristics in the second step …

*2022/10/1*

Jesper Møller

Aalborg Universitet

Translational psychiatry

##### Layer III pyramidal cells in the prefrontal cortex reveal morphological changes in subjects with depression, schizophrenia, and suicide

Brodmann Area 46 (BA46) has long been regarded as a hotspot of disease pathology in individuals with schizophrenia (SCH) and major depressive disorder (MDD). Pyramidal neurons in layer III of the Brodmann Area 46 (BA46) project to other cortical regions and play a fundamental role in corticocortical and thalamocortical circuits. The AutoCUTS-LM pipeline was used to study the 3-dimensional structural morphology and spatial organization of pyramidal cells. Using quantitative light microscopy, we used stereology to calculate the entire volume of layer III in BA46 and the total number and density of pyramidal cells. Volume tensors estimated by the planar rotator quantified the volume, shape, and nucleus displacement of pyramidal cells. All of these assessments were carried out in four groups of subjects: controls (C, n = 10), SCH (n = 10), MDD (n = 8), and suicide subjects with a history of depression (SU …

*2022/9/5*

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Aalborg Universitet

Graphs and Combinatorics

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*2022/6*

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Aalborg Universitet

Journal of Computational and Graphical Statistics

##### MCMC computations for Bayesian mixture models using repulsive point processes

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*2022/4/3*

Jesper Møller

Aalborg Universitet

Scandinavian Journal of Statistics

##### Approximate Bayesian inference for a spatial point process model exhibiting regularity and random aggregation

In this article, we propose a doubly stochastic spatial point process model with both aggregation and repulsion. This model combines the ideas behind Strauss processes and log Gaussian Cox processes. The likelihood for this model is not expressible in closed form but it is easy to simulate realizations under the model. We therefore explain how to use approximate Bayesian computation (ABC) to carry out statistical inference for this model. We suggest a method for model validation based on posterior predictions and global envelopes. We illustrate the ABC procedure and model validation approach using both simulated point patterns and a real data example.

*2022/3*

Jesper Møller

Aalborg Universitet

Journal of Algebraic Combinatorics

##### Equivariant Euler characteristics of unitary buildings

The (p-primary) equivariant Euler characteristics of the buildings for the general unitary groups over finite fields are determined.

*2021/11*

Jesper Møller

Aalborg Universitet

Communications Biology

##### Cellular 3D-reconstruction and analysis in the human cerebral cortex using automatic serial sections

Techniques involving three-dimensional (3D) tissue structure reconstruction and analysis provide a better understanding of changes in molecules and function. We have developed AutoCUTS-LM, an automated system that allows the latest advances in 3D tissue reconstruction and cellular analysis developments using light microscopy on various tissues, including archived tissue. The workflow in this paper involved advanced tissue sampling methods of the human cerebral cortex, an automated serial section collection system, digital tissue library, cell detection using convolution neural network, 3D cell reconstruction, and advanced analysis. Our results demonstrated the detailed structure of pyramidal cells (number, volume, diameter, sphericity and orientation) and their 3D spatial organization are arranged in a columnar structure. The pipeline of these combined techniques provides a detailed analysis of tissues …

*2021/9/2*

Jesper Møller

Aalborg Universitet

AMERICAN MATHEMATICAL SOCIETY

##### THE NUMBER OF p-ELEMENTS IN FINITE GROUPS OF LIE TYPE OF CHARACTERISTIC p

The combinatorics of the poset of p-radical p-subgroups of a finite group is used to count the number of p-elements.

*2021/7/1*

### Other articles from arXiv preprint arXiv:2404.09525 journal

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*