Negative dependence in tournaments has received attention in the literature. The property of negative orthant dependence (NOD) was proved for different tournament models with a special proof for each model. For general round-robin tournaments and knockout tournaments with random draws, Malinovsky and Rinott (2023) unified and simplified many existing results in the literature by proving a stronger property, negative association (NA). For a knockout tournament with a non-random draw, they presented an example to illustrate that ${\boldsymbol{S}}$ is NOD but not NA. However, their proof is not correct. In this paper, we establish the properties of negative regression dependence (NRD), negative left-tail dependence (NLTD), and negative right-tail dependence (NRTD) for a knockout tournament with a random draw and with players being of equal strength. For a knockout tournament with a non-random draw and with equal strength, we prove that ${\boldsymbol{S}}$ is NA and NRTD, while ${\boldsymbol{S}}$ is, in general, not NRD or NLTD.

Any margin of the multinomial distribution is multinomially distributed. Retaining this closure property, a family of generalized multinomial distributions is proposed. This family is characterized within multiplicative probability measures, using the Bell polynomial. The retained closure property simplifies marginal properties such as moments. The family can be obtained by conditioning independent infinitely divisible distributions on the total and also by mixing the multinomial distribution with the normalized infinitely divisible distribution. The closure property justifies a stochastic process of the family by Kolmogorov’s extension theorem. Over time, Gibbs partitions of a positive integer appear as the limiting distributions of the family.

Let $\{X_{i}\}_{i\geq1}$ be a sequence of independent and identically distributed random variables and $T\in\{1,2,\ldots\}$ a stopping time associated with this sequence. In this paper, the distribution of the minimum observation, $\min\{X_{1},X_{2},\ldots,X_{T}\}$, until the stopping time T is provided by proposing a methodology based on an appropriate change of the initial probability measure of the probability space to a truncated (shifted) one on the $X_{i}$. As an application of the aforementioned general result, the random variables $X_{1},X_{2},\ldots$ are considered to be the interarrival times (spacings) between successive appearances of events in a renewal counting process $\{Y_{t},t\geq0\}$, while the stopping time T is set to be the number of summands until the sum of the $X_{i}$ exceeds t for the first time, i.e. $T=Y_{t}+1$. Under this setup, the distribution of the minimal spacing, $D_{t}=\min\{X_{1},X_{2},\ldots,X_{Y_{t}+1}\}$, that starts in the interval [0, t] is investigated and a stochastic ordering relation for $D_{t}$ is obtained. In addition, bounds for the tail probability of $D_{t}$ are provided when the interarrival times have the increasing failure rate / decreasing failure rate property. In the special case of a Poisson process, an exact formula, as well as closed-form bounds and an asymptotic result, are derived for the tail probability of $D_{t}$. Furthermore, for renewal processes with Erlang and uniformly distributed interarrival times, exact and approximation formulae for the tail probability of $D_{t}$ are also proposed. Finally, numerical examples are presented to illustrate the aforementioned exact and asymptotic results, and practical applications are briefly discussed.

Let $X_H$ be the number of copies of a fixed graph H in G(n,p). In 2016, Gilmer and Kopparty conjectured that a local central limit theorem should hold for $X_H$ as long as H is connected, $p\gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$, where m(H) denotes the m-density of H. Recently, Sah and Sawhney showed that the Gilmer–Kopparty conjecture holds for constant p. In this paper, we show that the Gilmer–Kopparty conjecture holds for triangle counts in the sparse range. More precisely, if $p \in (4n^{-1/2}, 1/2)$, then

\[ {} {} {}\sup_{x\in \mathcal{L}}\left| \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* = x)\right|=n^{-1/2+o(1)}p^{1/2}, {} {}\]

$\sigma^2 = \mathbb{V}\text{ar}(X_{K_3})$

$X^{*}=(X_{K_3}-\mathbb{E}(X_{K_3}))/\sigma$

$\mathcal{L}$

$X^*$

$n^{-1}\ll p \lt c$

$c\in (0,1)$

$\ell_1$

$m_2$

We investigate the possibility of defining meaningful upper and lower quantization dimensions for a compactly supported Borel probability measure of order r, including negative values of r. To this end, we employ the concept of partition functions, which generalises the notion of the $L^q$-spectrum, thus extending the authors’ earlier work with Sanguo Zhu in a natural way. In particular, we derive inherent fractal-geometric bounds and easily verifiable necessary conditions for the existence of quantization dimensions. We state the exact asymptotics of the quantization error of negative order for absolutely continuous measures, thereby providing an affirmative answer to an open question regarding the geometric mean error posed by Graf and Luschgy in this journal in 2004.

This paper introduces the general ideas for parametric integral stochastic orders, with which a continuum of parametric functions are defined as a bridge between different classes of non-parametric functions. This approach allows one to identify a parametric function class over which two given random variables may violate the non-parametric stochastic order with specific patterns. The parameter used to name the parametric function class also measures the ratio of dominance violation for the corresponding non-parametric stochastic orders. Our framework, expanding the domain of stochastic orders, covers the existing studies of almost stochastic dominance. This leads to intuitive explanations and simpler proofs of existing results and their extensions.

Existing multivariate versions of the logistic probability distribution generally lack some of the useful properties of the univariate logistic distribution, such as its bounded score function or the tractability of its density function, or lack the rotational symmetry necessary for many applications. This paper clarifies some of the properties of such distributions and proposes a multivariate distribution closely related to the univariate logistic that has a tractable density, including the necessary normalising constant, bounded score function and elliptical symmetry. Some properties of its marginal distributions are explored, particularly in the bivariate case.

We define the generalized equilibrium distribution, that is the equilibrium distribution of a random variable with support in $\mathbb{R}$. This concept allows us to prove a new probabilistic generalization of Taylor’s theorem. Then, the generalized equilibrium distribution of two ordered random variables is considered and a probabilistic analog of the mean value theorem is proved. Results regarding distortion-based models and mean-median-mode relations are illustrated as well. Conditions for the unimodality of such distributions are obtained. We show that various stochastic orders and aging classes are preserved through the proposed equilibrium transformations. Further applications are provided in actuarial science, aiming to employ the new unimodal equilibrium distributions for some risk measures, such as Value-at-Risk and Conditional Tail Expectation.

Exchangeable partitions of the integers and their corresponding mass partitions on $\mathcal{P}_{\infty} = \{\mathbf{s} = (s_{1},s_{2},\ldots)\colon s_{1} \geq s_{2} \geq \cdots \geq 0$ and $\sum_{k=1}^{\infty}s_{k} = 1\}$ play a vital role in combinatorial stochastic processes and their applications. In this work, we continue our focus on the class of Gibbs partitions of the integers and the corresponding stable Poisson–Kingman-distributed mass partitions generated by the normalized jumps of a stable subordinator with an index $\alpha \in (0,1)$, subject to further mixing. This remarkable class of infinitely exchangeable random partitions is characterized by probabilities that have Gibbs (product) form. These partitions have practical applications in combinatorial stochastic processes, random tree/graph growth models, and Bayesian statistics. The most notable class consists of random partitions generated from the two-parameter Poisson–Dirichlet distribution $\mathrm{PD}(\alpha,\theta)$. While the utility of Gibbs partitions is recognized, there is limited understanding of the broader class. Here, as a continuation of our work, we address this gap by extending the dual coagulation/fragmentation results of Pitman (1999), developed for the Poisson–Dirichlet family, to all Gibbs models and their corresponding Poisson–Kingman mass partitions, creating nested families of Gibbs partitions and mass partitions. We focus primarily on fragmentation operations, identifying which classes correspond to these operations and providing significant calculations for the resulting Gibbs partitions. Furthermore, for completion, we provide definitive results for dual coagulation operations using dependent processes. We demonstrate the applicability of our results by establishing new findings for Brownian excursion partitions, Mittag-Leffler, and size-biased generalized gamma models.

Sharp, nonasymptotic bounds are obtained for the relative entropy between the distributions of sampling with and without replacement from an urn with balls of $c\geq 2$ colors. Our bounds are asymptotically tight in certain regimes and, unlike previous results, they depend on the number of balls of each color in the urn. The connection of these results with finite de Finetti-style theorems is explored, and it is observed that a sampling bound due to Stam (1978) combined with the convexity of relative entropy yield a new finite de Finetti bound in relative entropy, which achieves the optimal asymptotic convergence rate.

In this paper we adopt the probabilistic mean value theorem in order to study differences of the variances of transformed and stochastically ordered random variables, based on a suitable extension of the equilibrium operator. We also develop a rigorous approach aimed at expressing the variance of transformed random variables. This is based on a joint distribution which, in turn, involves the variance of the original random variable, as well as its mean residual lifetime and mean inactivity time. Then we provide applications to the additive hazards model and to some well-known random variables of interest in actuarial science. These deal with a new notion, called the ‘centred mean residual lifetime’, and a suitably related stochastic order. Finally, we also address the analysis of the differences of the variances of transformed discrete random variables thanks to the use of a discrete version of the equilibrium operator.

The present paper develops a unified approach when dealing with short- or long-range dependent processes with finite or infinite variance. We are concerned with the convergence rate in the strong law of large numbers (SLLN). Our main result is a Marcinkiewicz–Zygmund law of large numbers for $S_{n}(f)= \sum_{i=1}^{n}f(X_{i})$, where $\{X_i\}_{i\geq 1}$ is a real stationary Gaussian sequence and $f(\!\cdot\!)$ is a measurable function. Key technical tools in the proofs are new maximal inequalities for partial sums, which may be useful in other problems. Our results are obtained by employing truncation alongside new maximal inequalities. The result can help to differentiate the effects of long memory and heavy tails on the convergence rate for limit theorems.

We consider a stochastic model, called the replicator coalescent, describing a system of blocks of k different types that undergo pairwise mergers at rates depending on the block types: with rate $C_{ij}\geq 0$ blocks of type i and j merge, resulting in a single block of type i. The replicator coalescent can be seen as a generalisation of Kingman’s coalescent death chain in a multi-type setting, although without an underpinning exchangeable partition structure. The name is derived from a remarkable connection between the instantaneous dynamics of this multi-type coalescent when issued from an arbitrarily large number of blocks, and the so-called replicator equations from evolutionary game theory. By dilating time arbitrarily close to zero, we see that initially, on coming down from infinity, the replicator coalescent behaves like the solution to a certain replicator equation. Thereafter, stochastic effects are felt and the process evolves more in the spirit of a multi-type death chain.

Competing and complementary risk (CCR) problems are often modelled using a class of distributions of the maximum, or minimum, of a random number of independent and identically distributed random variables, called the CCR class of distributions. While CCR distributions generally do not have an easy-to-calculate density or probability mass function, two special cases, namely the Poisson–exponential and exponential–geometric distributions, can easily be calculated. Hence, it is of interest to approximate CCR distributions with these simpler distributions. In this paper, we develop Stein’s method for the CCR class of distributions to provide a general comparison method for bounding the distance between two CCR distributions, and we contrast this approach with bounds obtained using a Lindeberg argument. We detail the comparisons for Poisson–exponential, and exponential–geometric distributions.

We show that $\alpha $-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for $0<\alpha <1$ and every $\alpha $-stable Lévy motion ${\mathbb {W}}$, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1\leq \alpha <2$ and every symmetric $\alpha $-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to ${\mathbb {W}}$; for $1< \alpha <2$ and every $-1\leq \beta \leq 1$ there exists a function f whose associated time series is in the classical domain of attraction of an $S_\alpha (\ln (2), \beta ,0)$ random variable.

Motivated by the investigation of probability distributions with finite variance but heavy tails, we study infinitely divisible laws whose Lévy measure is characterized by a radial component of geometric (tempered) stable type. We closely investigate the univariate case: characteristic exponents and cumulants are calculated, as well as spectral densities; absolute continuity relations are shown, and short- and long-time scaling limits of the associated Lévy processes analyzed. Finally, we derive some properties of the involved probability density functions.

We introduce a comprehensive method for establishing stochastic orders among order statistics in the independent and identically distributed case. This approach relies on the assumption that the underlying distribution is linked to a reference distribution through a transform order. Notably, this method exhibits broad applicability, particularly since several well-known nonparametric distribution families can be defined using relevant transform orders, including the convex and the star transform orders. Moreover, for convex-ordered families, we show that an application of Jensen’s inequality gives bounds for the probability that a random variable exceeds the expected value of its corresponding order statistic.

Conditional risk measures and their associated risk contribution measures are commonly employed in finance and actuarial science for evaluating systemic risk and quantifying the effects of risk interactions. This paper introduces various types of contribution ratio measures based on the multivariate conditional value-at-risk (MCoVaR), multivariate conditional expected shortfall (MCoES), and multivariate marginal mean excess (MMME) studied in [34] (Ortega-Jiménez, P., Sordo, M., & Suárez-Llorens, A. (2021). Stochastic orders and multivariate measures of risk contagion. Insurance: Mathematics and Economics, vol. 96, 199–207) and [11] (Das, B., & Fasen-Hartmann, V. (2018). Risk contagion under regular variation and asymptotic tail independence. Journal of Multivariate Analysis 165(1), 194–215) to assess the relative effects of a single risk when other risks in a group are in distress. The properties of these contribution risk measures are examined, and sufficient conditions for comparing these measures between two sets of random vectors are established using univariate and multivariate stochastic orders and statistically dependent notions. Numerical examples are presented to validate these conditions. Finally, a real dataset from the cryptocurrency market is used to analyze the spillover effects through our proposed contribution measures.

For a continuous-time phase-type (PH) distribution, starting with its Laplace–Stieltjes transform, we obtain a necessary and sufficient condition for its minimal PH representation to have the same order as its algebraic degree. To facilitate finding this minimal representation, we transform this condition equivalently into a non-convex optimization problem, which can be effectively addressed using an alternating minimization algorithm. The algorithm convergence is also proved. Moreover, the method we develop for the continuous-time PH distributions can be used directly for the discrete-time PH distributions after establishing an equivalence between the minimal representation problems for continuous-time and discrete-time PH distributions.

We propose Rényi information generating function (RIGF) and discuss its properties. A connection between the RIGF and the diversity index is proposed for discrete-type random variables. The relation between the RIGF and Shannon entropy of order q > 0 is established and several bounds are obtained. The RIGF of escort distribution is derived. Furthermore, we introduce the Rényi divergence information generating function (RDIGF) and discuss its effect under monotone transformations. We present nonparametric and parametric estimators of the RIGF. A simulation study is carried out and a real data relating to the failure times of electronic components is analyzed. A comparison study between the nonparametric and parametric estimators is made in terms of the standard deviation, absolute bias, and mean square error. We have observed superior performance for the newly proposed estimators. Some applications of the proposed RIGF and RDIGF are provided. For three coherent systems, we calculate the values of the RIGF and other well-established uncertainty measures, and similar behavior of the RIGF is observed. Further, a study regarding the usefulness of the RDIGF and RIGF as model selection criteria is conducted. Finally, three chaotic maps are considered and then used to establish a validation of the proposed information generating function.