Preface to the Special Issue on “Probability, Statistics and Random Processes”

1. Introduction

Probability, statistics, and the modeling of random processes are well-established yet actively developing domains of science, with numerous contemporary applications in finance, economics, engineering, physics, biology, and many other areas. Many natural and anthropogenic phenomena are governed by stochastic laws, and in modern computation and data analysis, both frequentist and Bayesian statistical methods play fundamental roles in solving problems in hypothesis testing, regression modeling, sensitivity analysis, and more. In these fields, researchers continue to introduce new theoretical developments (for example, novel probability distributions or convergence results) and address complex applied problems across disciplines.

This Special Issue was launched to highlight recent theoretical and applied advances in probability, statistics, and stochastic processes. The call for papers welcomed high-quality contributions providing pioneering results, solving challenging problems, or proposing novel and advanced methods and techniques. The response from the community was notable, and after a rigorous peer-review process, nine papers were accepted for publication. These contributions span a broad range of topics—from fundamental probability theory and statistical inference to applications in finance and beyond—demonstrating the vibrancy and importance of research in this domain.

2. A Summary of the Papers

The first paper, written by Yang et al. (Contribution 1), examines the Wasserstein distance between discrete and continuous empirical distributions. Specifically, the authors consider n i.i.d. random points uniformly distributed within a d-dimensional ball and study their empirical probability measure versus the uniform measure on the ball. They derive asymptotic estimates for the expected Wasserstein metric as n approaches infinity, providing an explicit convergence rate. Furthermore, Yang et al. extend the analysis to a mixed (spatial Poisson process) framework where the number of sample points is itself random, thereby bridging the gap between discrete random samples and continuous distributions using optimal transport metrics.

Li (Contribution 2) focuses on birth–death processes that are subject to two different types of catastrophic events. An elegant analytic result is obtained: the Laplace transform of the transition probability function for a birth–death process with two types of catastrophes is expressed in a closed form using the known Laplace transform of the transition probabilities for a corresponding birth–death process without catastrophes. Li then considers the distribution of the times at which the first catastrophe occurs. The Laplace transform of this hitting time’s probability density is derived, and from it the author obtains explicit formulas for the expected value and variance of the time at which the catastrophe occurs. These results provide deeper insight into the impact of rare catastrophic events on the dynamics of birth–death stochastic processes.

Varypaev (Contribution 3) addresses a challenging estimation problem in a multidimensional linear system model with nuisance parameters. The author considers a linear system with one input (regressor) and m outputs, where the input sequence consists of unknown deterministic parameters (interpreted as an ‘infinite number of nuisance parameters’). Only the m output signals, corrupted by additive stationary Gaussian noise, can be observed. Assuming local asymptotic normality for the likelihood of their observation, the paper establishes the consistency (convergence to the true value) of a likelihood-based estimator for the system’s parameters. Moreover, Varypaev derives the asymptotic form of the estimator’s covariance matrix in the limit as the number of observations tends to infinity, thus generalizing classic results in system identification to a case with infinite nuisance parameters.

Zhang et al. (Contribution 4) develop a new statistical test for detecting persistence changes in time series with heavy-tailed noise (observations with infinite variance). The authors propose a test statistic to identify whether a time series switches from stationary to nonstationary behavior or vice versa, even when the innovations follow a power law distribution (stable law) with infinite second moments. They derive the asymptotic distribution of the test statistic under the null hypothesis that there is no change in the persistence, showing it to be a functional of a stable Lévy process. To address the null distribution’s dependence on an unknown tail index, a modified block bootstrap procedure is employed to obtain critical values for the test. Via simulations, Zhang et al. demonstrate that the bootstrap-based test has good size and power properties and does not spuriously detect changes when none are present (avoiding ‘over-rejection’). Notably, the test can also infer the direction of a persistence change (distinguishing stationary-to-nonstationary shifts from the opposite). Using real data, the authors apply their method to the U.S. inflation rate and USD/CNY exchange rate series; in each case, they find statistically significant evidence of a change in persistence, illustrating the practical relevance of the proposed test.

Mihova et al. (Contribution 5) present a study on models for forecasting stock prices under the conditions of a financial crisis. The authors compare two different modeling approaches for predicting stock prices: (i) classical time series models, specifically ARIMA models, and (ii) a modified ordinary differential equation (ODE)-based model previously developed by some of the authors. Using data on four Bulgarian companies’ stock prices, the paper evaluates how well each model can forecast one-period-ahead returns during a period of turbulent market conditions (characterized as a financial crisis). The returns predicted by both the ARIMA and ODE-based models are then used to construct optimal investment portfolios using an enhanced Markowitz’s mean variance framework. Mihova et al. assess the forecasting errors of each model and compare the resulting portfolios in terms of their composition, risk, and expected return. This validation study provides insight into the reliability of different forecasting approaches under volatile market conditions and their implications for portfolio optimization: for instance, differences in models’ accuracy can translate into different portfolio allocations and risk–return profiles. The results help to highlight modeling techniques that may be more robust or informative during periods of financial instability.

Considering geometric probability, Rodrigo et al. (Contribution 6) revisit the classical Bertrand paradox and offer two novel approaches to compute the probability in Bertrand’s chord problem. The problem asks, if a chord of a circle is chosen at random, what is the probability that the chord is longer than the side of an inscribed equilateral triangle? Bertrand famously showed that the answer could vary (e.g., it could be ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$, ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$, or ${\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$) depending on the method of random chord selection, thus posing a paradox in defining ‘randomness’ for chords. In this paper, Rodrigo et al. provide two new methods to obtain a solution. The first employs an immersion of the problem in ${\mathbb{R}}^4$, using a higher-dimensional geometric construction to ensure unambiguous random chord selection. The second method defines a direct probability measure on the space of chords of a circle to avoid any ambiguity in the selection procedure. Both approaches yield a well-defined probability for the event of interest (the selection of a chord longer than the triangle’s side) and shed further light on the nature of Bertrand’s paradox by proposing alternative ways to interpret ‘random’ chords. These fresh perspectives contribute to the long-standing discussion on and didactics of random geometric selection.

Alhihi and Almheidat (Contribution 7) investigate the use of statistical inference to determine Pianka’s overlap coefficient in the case of two exponential distributions. Overlap measures such as Pianka’s coefficient are used in ecology and other fields to quantify the similarity or overlap between two probability distributions (or populations) based on their density functions. In this paper, the authors derive formulas for Pianka’s overlap coefficient when both populations have exponential distributions and develop methods to estimate this coefficient from data. They obtain the maximum likelihood estimator (MLE) for Pianka’s overlap and also propose a Bayesian estimator, assuming that suitable prior information is available. Through a simulation study, Alhihi and Almheidat evaluate the bias and mean squared error of these estimators, illustrating their finite-sample performance. They also construct confidence intervals for the true overlap coefficient, providing practitioners with tools to draw inferences about population overlaps. The findings indicate how one can reliably estimate and quantify the overlap between exponential populations, and the comparative analysis of the MLE and Bayesian approaches offers guidance on their use under different sample sizes or prior assumptions.

Chechile (Contribution 8) provides a comprehensive resolution to Bertrand’s paradox and examines its implications for a related problem in Bayesian inference known as the Bing–Fisher problem. As discussed earlier, Bertrand’s paradox, a famous problem in probability theory, arises from the fact that applying different ‘reasonable’ methods to choose a random chord of a circle results in different probabilities that the chord will be longer than the side of an inscribed equilateral triangle. In this paper, the paradox is resolved by carefully specifying the underlying random variable and constraints: Chechile clarifies the primary variate to which the principle of maximum entropy should be applied and imposes mathematically derived constraints on the random process when performing nonlinear transformations to secondary variables. These steps eliminate ambiguity and result in a unique solution to Bertrand’s problem, one that diverges from the classical solutions proposed by Bertrand and others. The author shows that the previous solutions corresponded to sampling procedures that were not truly ‘random’ in an unconstrained sense. Furthermore, Chechile applies the same conceptual framework to the Bing–Fisher problem (which concerns the selection of an appropriate prior distribution in Bayesian inference, which has historically yielded inconsistent results). The resolution of the Bing–Fisher problem presented in this work rebuts certain philosophical arguments against the Bayesian approach to inference that were grounded in the apparent inconsistencies these paradoxes exemplify. By resolving both Bertrand’s paradox and the Bing–Fisher problem, the paper strengthens the case for using a Bayesian approach and addresses key issues facing the foundations of probability and statistics.

Akoto and Mexia (Contribution 9) explore the concept of consistency in statistical decision theory within the context of multinomial models. In classical statistics, consistency typically refers to an estimator converging in probability to the true parameter value as the sample size n grows. Akoto and Mexia extend this idea to decision-making problems: they consider a scenario where we have to choose among multiple actions (decisions) in a setting modeled by a multinomial distribution (which can have either a finite or countably infinite number of outcome categories). Each decision is associated with a cost, which depends on the true state (parameter) and chosen action. The authors find that under quite general conditions, the probability of making the optimal decision (the one with the minimum expected cost) approaches 1 as $n\to \infty$. In other words, the decision rule is consistent in the sense that, given enough data, it almost certainly results in the correct action with a minimal risk. The authors demonstrate that this result holds for both finite multinomial models and those with an infinite (numerable) set of categories. They show, for instance, that the empirical frequency estimator of the category probabilities is a consistent estimator of the true probability vector, and using this estimator in the decision rule yields consistency in the decision itself. This work introduces a broader notion of consistency that connects estimation and decision-making: it assures us that, in large samples, not only do parameter estimates converge, but the actual decisions based on those estimates also converge to the best possible decision. This is a valuable contribution to statistical decision theory, highlighting the conditions under which the decision-making process essentially becomes reliable with large data sets.

3. Conclusions

As the Guest Editor of this Special Issue, I am very pleased with the breadth and quality of the contributions. I would like to sincerely thank all the authors for submitting their outstanding research and their collaboration during the revision process. I am also grateful to the expert reviewers who dedicated their time and expertise to providing valuable feedback and constructive criticism, helping to maintain the high scientific standards of this Special Issue. Special thanks are due to the editorial team of Mathematics at MDPI for the professional support they provided and their efficient management of the review process, which greatly facilitated the publication of this Special Issue.

The overarching goal of this Special Issue, “Probability, Statistics and Random Processes”, is to showcase contemporary advancements and foster cross-pollination between theory and practice in these interrelated fields. We believe that the selected papers achieve this goal, highlighting both cutting-edge theoretical developments and impactful applications. It is our hope that this collection of articles will serve as a useful resource for researchers, inspire further investigations, and stimulate new ideas and collaboration in the probability and statistics community.