Abstract We derive bounds for the Shannon entropy and the extropy of the largest order statistic and the associated normalized sequence in the extreme value theory setting. The largest order statistic comes from a set of n independent and identically distributed random variables of finite or infinite ( n right arrow normal infinity n → ∞ $n\rightarrow \infty $ ) number, governed by a log-concave distribution. A characterization of the exponential distribution is provided as the model that maximizes the Shannon entropy and the extropy of the largest order statistic and the associated normalized maximum value in a large sample size regime. This characterization is exploited to provide an alternative, immediate proof of the convergence of the Shannon entropy and the extropy of the normalized maxima to the respective measures for the Gumbel distribution, studied recently for Shannon entropy by Johnson ((2024). J. Appl. Probab. 61 , 244–254.).
Abstract The study of random networks generally requires that vertices be independently and uniformly distributed over disjoint regions such as a Poisson point process. In this work, we venture beyond this standard paradigm and investigate a stochastic forest obtained from a drainage network model constructed on a random subset of double struck upper Z squared Z 2 $\mathbb{Z}^2$ , where points in double struck upper Z squared Z 2 $\mathbb{Z}^2$ are randomly perturbed such that both horizontal and vertical perturbations are given by exponentially decaying discrete random variables with unbounded support and edges between two such points are specified according to the distance between them. We restrict vertical perturbations in the upward direction only. Here we establish that (i) the resultant stochastic network is a single tree almost surely and (ii) as a collection of paths, under diffusive scaling the resulting network converges to the Brownian web.
Abstract In recent years, there has been effort to extend the classical notion of phylogenetic balance, originally defined in the context of trees, to networks. One of the most natural ways to do this is with the so-called upper B 2 B 2 $B_2$ index. In this paper, we study the upper B 2 B 2 $B_2$ index for a prominent class of phylogenetic networks: galled trees. We show that the upper B 2 B 2 $B_2$ index of a uniform leaf-labeled galled tree converges in distribution as the network becomes large. We characterize the corresponding limiting distribution, and provide a way to compute its moments. This is the first time that a balance index has been studied to this level of detail for a random phylogenetic network. One specificity of this work is that we use two different and independent approaches, each with its advantages: analytic combinatorics, and local limits. The analytic combinatorics approach is more direct, as it relies on standard tools, but it involves slightly more complex calculations. Because it has not previously been used to study such questions, the local limit approach requires the development of an extensive framework beforehand; however, this framework is interesting in itself, and can be used to tackle other, similar problems.
Abstract We study a model of social learning on rooted regular trees. An agent is stationed at each vertex of double struck upper T Subscript m T m $\mathbb{T}_{m}$ , the rooted tree in which each vertex has precisely m children, and at any time step t element of double struck upper N 0 t ∈ N 0 $t \in \mathbb{N}_{0}$ , the agent is allowed to select one of two available technologies: B and R . Let the technology chosen by the agent at vertex v of double struck upper T Subscript m T m $\mathbb{T}_{m}$ , at time step t , be upper C Subscript t Baseline left parenthesis v right parenthesis C t ( v ) $C_{t}(v)$ . We begin with the independent and identically distributed (i.i.d.) collection StartSet upper C 0 left parenthesis v right parenthesis colon v element of double struck upper T Subscript m Baseline EndSet { C 0 ( v ) : v ∈ T m } $\{C_{0}(v)\,:\, v \in \mathbb{T}_{m}\}$ , where upper C 0 left parenthesis v right parenthesis equals upper B C 0 ( v ) = B $C_{0}(v)=B$ with probability pi 0 π 0 $\pi_{0}$ . During the epoch t , the agent at vertex v performs an experiment that results in success with probability p Subscript upper B p B $p_{B}$ if upper C Subscript t Baseline left parenthesis v right parenthesis equals upper B C t ( v ) = B $C_{t}(v)=B$ , and with probability p Subscript upper R p R $p_{R}$ if upper C Subscript t Baseline left parenthesis v right parenthesis equals upper R C t ( v ) = R $C_{t}(v)=R$ . If the children of v are denoted v 1 comma ellipsis comma v Subscript m Baseline v 1 , … , v m $v_{1}, \ldots, v_{m}$ , the agent at v updates their technology to upper C Subscript t plus 1 Baseline left parenthesis v right parenthesis equals upper B C t + 1 ( v ) = B $C_{t+1}(v)=B$ if the number of successes among all v Subscript i v i $v_{i}$ (where i element of StartSet 1 comma 2 comma ellipsis comma m EndSet i ∈ { 1 , 2 , … , m } $i \in \{1,2,\ldots,m\}$ ) with upper C Subscript t Baseline left parenthesis v Subscript i Baseline right parenthesis equals upper B C t ( v i ) = B $C_{t}(v_{i})=B$ exceeds, strictly, the number of successes among all v Subscript j v j $v_{j}$ (where j element of StartSet 1 comma 2 comma ellipsis comma m EndSet j ∈ { 1 , 2 , … , m } $j \in \{1,2,\ldots,m\}$ ) with upper C Subscript t Baseline left parenthesis v Subscript j Baseline right parenthesis equals upper R C t ( v j ) = R $C_{t}(v_{j})=R$ . If these two numbers are equal then the agent at v sets upper C Subscript t plus 1 Baseline left parenthesis v right parenthesis equals upper B C t + 1 ( v ) = B $C_{t+1}(v)=B$ with probability 1 divided by 2 1 / 2 $1/2$ . In all other cases, upper C Subscript t plus 1 Baseline left parenthesis v right parenthesis equals upper R C t + 1 ( v ) = R $C_{t+1}(v)=R$ . We show that StartSet upper C Subscript t Baseline left parenthesis v right parenthesis colon v element of double struck upper T Subscript m Baseline EndSet { C t ( v ) : v ∈ T m } $\{C_{t}(v)\,:\, v \in \mathbb{T}_{m}\}$ is i.i.d. as well, with upper C Subscript t Baseline left parenthesis v right parenthesis equals upper B C t ( v ) = B $C_{t}(v)=B$ with probability pi Subscript t π t $\pi_{t}$ , where the sequence left brace pi Subscript t Baseline right brace Subscript t element of double struck upper N 0 { π t } t ∈ N 0 $\{\pi_{t}\}_{t \in \mathbb{N}_{0}}$ converges to a fixed point pi π $\pi$ , in [0, 1], of a function g Subscript m g m $g_{m}$ . We show that for m greater than or slanted equals 3 m ⩾ 3 $m \geqslant 3$ , there exists a p left parenthesis m right parenthesis element of left parenthesis 0 comma 1 right parenthesis p ( m ) ∈ ( 0 , 1 ) $p(m) \in (0,1)$ such that g Subscript m g m $g_{m}$ has the unique fixed point 1 divided by 2 1 / 2 $1/2$ when p less than or slanted equals p left parenthesis m right parenthesis p ⩽ p ( m ) $p \leqslant p(m)$ , and three distinct fixed points, of the form alpha α $\alpha$ , 1 divided by 2 1 / 2 $1/2$ , and 1 minus alpha 1 − α $1-\alpha$ , for some alpha element of left bracket 0 comma 1 divided by 2 right parenthesis α ∈ [ 0 , 1 / 2 ) $\alpha \in [0,1/2)$ when p greater than p left parenthesis m right parenthesis p > p ( m ) $p > p(m)$ . When m equals 3 m = 3 $m=3$ , p Subscript upper B Baseline equals 1 p B = 1 $p_{B}=1$ , and p Subscript upper R Baseline element of left bracket 0 comma 1 right parenthesis p R ∈ [ 0 , 1 ) $p_{R} \in [0,1)$ , we show that the function g 3 g 3 $g_{3}$ (i) has a unique fixed point, 1, when p Subscript upper R Baseline less than StartRoot 3 EndRoot minus 1 p R < 3 − 1 $p_{R} < \sqrt{3}-1$ , (ii) has two distinct fixed points, one of which is 1, when p Subscript upper R Baseline equals StartRoot 3 EndRoot minus 1 p R = 3 − 1 $p_{R} = \sqrt{3}-1$ , and (iii) has three distinct fixed points, one of which is 1, when p Subscript upper R Baseline greater than StartRoot 3 EndRoot minus 1 p R > 3 − 1 $p_{R} > \sqrt{3}-1$ . When g Subscript m g m $g_{m}$ has multiple fixed points, we also specify which of these fixed points pi π $\pi$ equals, depending on pi 0 π 0 $\pi_{0}$ . Finally, for m equals 2 m = 2 $m=2$ , we describe the behaviour of g 2 g 2 $g_{2}$ for all values of p Subscript upper B p B $p_{B}$ and p Subscript upper R p R $p_{R}$ .
This paper addresses a class of value-maximization problems in storage-type optimal control involving operational scale and withdrawals with absorption. Moving beyond the drifted Brownian motion model, we develop a unified methodological approach under a general linear diffusion framework. We demonstrate the solvability of a broad class of problems by deriving semi-explicit optimal strategies. We further establish conditions for the existence of fully explicit solutions and illustrate solution methods with examples based on models for a variety of scenarios, both where explicit solutions exist and where only semi-explicit solutions are available.
