Abstract The stochastic ordering of order statistics plays a central role in many fields involving probability and statistics. Over the past decade, significant progress has been made in ordering the first and second order statistics from heterogeneous or statistically dependent observations. This paper further develops necessary and sufficient conditions for the third-largest and third-smallest order statistics from observations linked by an Archimedean copula and following, respectively, the proportional hazards, proportional reverse hazards, and scale models. The characterization results and sufficient conditions presented herein not only enrich the existing literature but also provide novel theoretical insights for applications in fields such as reliability systems, multiple outlier models, and the design of third-price auction mechanisms.
Abstract Introduced by Müller et al. in their seminal paper (2017), fractional stochastic dominance (SD) offers a nuanced approach to ordering distributions. In this paper we propose a fundamentally new framework by replacing the fixed parameter gamma element of left bracket 0 comma 1 right bracket γ ∈ [ 0 , 1 ] $\gamma \in [0, 1]$ in fractional SD with a function bold gamma colon double struck upper R right arrow left bracket 0 comma 1 right bracket 𝛄 : R → [ 0 , 1 ] $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$ . This yields two novel families, multi-fractional stochastic dominance (MFSD) and functional fractional stochastic dominance (FFSD). They enable the ranking of a broader range of distributions and incorporate a more informative utility class, including those with local non-concavities whose steepness varies depending on the location. Furthermore, our framework introduces the concept of partial greediness, which dynamically captures how behaviour of decision makers adapts to changes in wealth. We also extend this framework to encompass almost stochastic dominance. We provide the mathematical foundations of our generalized framework and study how it offers a novel tool for ordering distributions across various settings.
Abstract We consider a random right-angled Coxeter group (RACG), upper W Subscript normal upper Gamma W Γ $W_{\Gamma}$ , whose presentation graph normal upper Gamma Γ $\Gamma$ is taken to be an Erdős–Rényi random graph, i.e. normal upper Gamma tilde script upper G Subscript n comma p Γ ∼ G n , p $\Gamma\sim\mathcal{G}_{n,p}$ . We use techniques from probabilistic combinatorics to establish several new results about the geometry of this random group model. Resolving a conjecture of Susse, we determine the connectivity threshold for square percolation on the random graph normal upper Gamma tilde script upper G Subscript n comma p Γ ∼ G n , p $\Gamma\sim\mathcal{G}_{n,p}$ . We use this result to determine a large range of p for which the random RACG upper W Subscript normal upper Gamma W Γ $W_{\Gamma}$ has a unique cubical coarse median structure. Until Fioravanti et al. (2026), there were no non-hyperbolic examples of groups with cubical coarse rigidity. Perhaps surprisingly, we show that the property of coarse cubical rigidity is in fact typically satisfied by a random RACG for a wide range of the parameter p , including p equals one half p = 1 2 $p=\frac12$ , for which the random RACG is also typically non-hyperbolic.
Abstract Let alpha Subscript n Baseline left parenthesis dot right parenthesis equals double struck upper P left parenthesis upper X Subscript n plus 1 Baseline element of dot vertical bar upper X 1 comma ellipsis comma upper X Subscript n Baseline right parenthesis α n ( ⋅ ) = P ( X n + 1 ∈ ⋅ ∣ X 1 , … , X n ) $\alpha_n({\cdot})=\mathbb{P}\bigl(X_{n+1}\in{\cdot}\mid X_1,\ldots,X_n\bigr)$ be the predictive distributions of a sequence left parenthesis upper X 1 comma upper X 2 comma ellipsis right parenthesis ( X 1 , X 2 , … ) $(X_1,X_2,\ldots)$ of p -dimensional random vectors. Suppose alpha Subscript n Baseline equals script upper N left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis α n = N ( M n , Q n ) $\alpha_n=\mathcal{N}(M_n,Q_n)$ , where upper M Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts upper X Subscript i M n = ( 1 / n ) ∑ i = 1 n X i $M_n=({1}/{n})\sum_{i=1}^nX_i$ and upper Q Subscript n Baseline equals left parenthesis 1 divided by n right parenthesis sigma summation Underscript i equals 1 Overscript n Endscripts left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis left parenthesis upper X Subscript i Baseline minus upper M Subscript n Baseline right parenthesis Superscript down tack Q n = ( 1 / n ) ∑ i = 1 n ( X i − M n ) ( X i − M n ) ⊤ $Q_n=({1}/{n})\sum_{i=1}^n(X_i-M_n)(X_i-M_n)^{\top}$ . Then there is a random probability measure alpha α $\alpha$ on the Borel subsets of double struck upper R Superscript p R p $\mathbb{R}^p$ such that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 ‖ α n − α ‖ ⟶ a . s . 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ , where StartMetric dot EndMetric ‖ ⋅ ‖ $\lVert{\cdot}\rVert$ is the total variation distance. An explicit expression for alpha α $\alpha$ is provided and the convergence rate of StartMetric alpha Subscript n Baseline minus alpha EndMetric ‖ α n − α ‖ $\lVert\alpha_n-\alpha\rVert$ is shown to be arbitrarily close to n Superscript negative 1 divided by 2 n − 1 / 2 $n^{-1/2}$ . Moreover, it is still true that StartMetric alpha Subscript n Baseline minus alpha EndMetric long right arrow Overscript normal a period normal s period Endscripts 0 ‖ α n − α ‖ ⟶ a . s . 0 $\lVert\alpha_n-\alpha\rVert\overset{\mathrm{a.s.}}\longrightarrow 0$ even if alpha Subscript n Baseline equals script upper L left parenthesis upper M Subscript n Baseline comma upper Q Subscript n Baseline right parenthesis α n = L ( M n , Q n ) $\alpha_n=\mathcal{L}(M_n,Q_n)$ , where script upper L L $\mathcal{L}$ belongs to a class of distributions much larger than the normal. The predictives alpha Subscript n α n $\alpha_n$ are useful in various frameworks, including Bayesian predictive inference and predictive resampling. Finally, the asymptotic behavior of copula-based predictive distributions is investigated and a numerical experiment is performed.
In this paper, we analyze the distribution of the total overlapping time spent with other customers in the M-lambda/M-& micro;/1 queue with First-Come First-Served service discipline. We show that the Laplace-Stieltjes transform of the overlapping time reduces to an incomplete gamma function representation. We also calculate the transform of the joint distribution of the overlapping time and the number of overlaps. In addition, we prove a heavy-traffic limit for the total overlapping time with scaling 1/(1-rho)(2) to a Weibull (1/2, 1/& micro;) random variable. Finally, for the M-lambda/G/1 case, we derive the first two moments of the overlapping time.
