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.
This paper focuses mainly on the Euler scheme of stochastic delay differential equations with locally Lipschitz coefficients. The convergence in probability of the Euler scheme and the corresponding weak limit process of the normalized error process are derived. Furthermore, this paper also considers a class of specific degenerate stochastic delay equations and obtains the associated weak limit process for the stronger error process. The error parameter of this stronger error process for such a degenerate system is n instead of $\sqrt{n}$ in the normalized error process. This causes substantial challenges in the analysis and proofs and the weak limit process also becomes more complicated and involves some additional terms. This result is new and interesting even for the non-delay case.
Past research has indicated that the covariance of the stochastic gradient descent (SGD) error done via minibatching plays a critical role in determining its regularization and escape from low potential points. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the SGD algorithm, with multiplicative noise, introduced in previous work (Wu et al (2016) Int. Conf. on Machine Learning, PMLR, pp. 10367-10376), which has a much more general noise class than the SGD algorithm done via minibatching. We establish non-asymptotic bounds for the multiplicative SGD algorithm in the Wasserstein distance. We also show that the error term for the algorithm is approximately a scaled Gaussian distribution with mean 0 at any fixed point.
This paper derives explicit expressions for drawdown-based two-sided exit identities involving the overshoots and undershoots at the exit times under Poisson observation times for spectrally negative L & eacute;vy risk processes by using fluctuation theory. All resulting Laplace transforms of the risk quantities of interest are expressed in terms of the scale functions of the spectrally negative L & eacute;vy processes.
We consider a highly generalized set-up in which each vertex of the infinite two-dimensional square lattice graph (whose set of vertices is $\mathbb{Z}<^>{2}$ , with each vertex (x, y) adjacent to each of $(x+1,y)$ and $(x,y+1)$ ) is assigned, independent of all else, a label that reads trap with probability p, target with probability q, and open with the remaining probability $(1-p-q)$ , and, in addition, each edge is assigned, independent of all else, a label that reads trap with probability r and open with probability $(1-r)$ . This model encompasses the seemingly more general model where, in addition to all the vertex-labels and edge-labels described above, an edge can also be labeled as a target, since assigning the label of target to an edge going from (x, y) to either $(x+1,y)$ or $(x,y+1)$ is equivalent to marking the vertex (x, y) as a trap. A percolation game is played on this random board, involving two players and a token. The players take turns to make moves, where a move involves relocating the token from where it is currently located, say some vertex $(x,y) \in \mathbb{Z}<^>{2}$ , to any one of $(x+1,y)$ and $(x,y+1)$ . A player wins if she is able to move the token to a vertex labeled as a target, or force her opponent to either move the token to a vertex labeled as a trap or along an edge labeled as a trap. We seek to find a regime, in terms of values of the parameters p, q, and r, in which the probability of this game resulting in a draw equals 0. We further consider special cases of this game, such as when each edge is assigned, independently, a label that reads trap with probability r, target with probability s, and open with probability $(1-r-s)$ , but the vertices are left unlabeled, and various regimes of values of r and s are explored in which the probability of draw is guaranteed to be 0. We show that the probability of draw in each such game equals 0 if and only if a suitably defined probabilistic cellular automaton (PCA) is ergodic, following which we implement the technique of weight functions or potential functions to investigate the regimes in which said PCA is ergodic. We mention here that one of the main results of Holroyd et al. (2019 Probab. Theory Related Fields 174, 1187-1217) follows as a special case of our main result. Moreover, our result shows that a phase transition happens at the origin (i.e. at $(p,q,r)=(0,0,0)$ in the case of generalized percolation games, and at $(r,s)=(0,0)$ in the case of bond percolation games) in the sense that, the probability of draw equals 1 at $(p,q,r)=(0,0,0)$ (respectively, at $(r,s)=(0,0)$ ), whereas in every neighborhood around (0, 0, 0) (respectively, (0, 0)), there exists some value of (p, q, r) (respectively, (r, s)) for which the probability of draw equals 0.
The marked Hawkes risk process is a compound point process where the occurrence and amplitude of past events impact the future. Since data in real life are acquired over a discrete time grid, we propose a strong discrete-time approximation of the continuous-time risk process obtained by embedding from the same Poisson measure. We then prove trajectorial convergence results in both fractional Sobolev spaces and the Skorokhod space, hence extending the theorems proven in Huang and Khabou ((2023). Stoch. Process. Appl. 161, 201-241) and Kirchner ((2016). Stoch. Process. Appl. 126(8), 2494-2525). We also provide upper bounds on the convergence speed with explicit dependence on the size of the discretization step, the time horizon, and the regularity of the kernel.
The relevation model is a fundamental tool in reliability engineering for assessing the effectiveness of redundancy allocation in coherent systems. In this study, we address the problem of allocation of relevations for one or two nodes in a coherent system with independent components to enhance system reliability. We establish results concerning the usual stochastic and hazard rate orders for coherent systems. Moreover, we illustrate our findings with a range of examples and counterexamples. In addition, we conduct a simulation-based study and a real data analysis to further illustrate the application of our results. Lastly, we study the case of the minimal repair policy in detail.
We prove new results about comparing the efficiency of general state space Markov chain Monte Carlo algorithms that randomly select a possibly different reversible method at each step (previously known only for finite state spaces). We also provide new, simpler, more accessible proofs of key results, and analyse numerous examples. We provide a full proof of the formula for the asymptotic variance for real-valued functionals on $\varphi$ -irreducible reversible Markov chains, first introduced by Kipnis and Varadhan (1986, Commun. Math. Phys. 104, 1-19). Given two Markov kernels P and Q with stationary measure $\pi$ , we say that the Markov kernel P efficiency-dominates the Markov kernel Q if the asymptotic variance with respect to P is at most the asymptotic variance with respect to Q for every real-valued functional $f\in L<^>2(\pi)$ . Assuming only a basic background in functional analysis, we prove that for two reversible Markov kernels P and Q, P efficiency-dominates Q if and only if the operator $\mathcal{Q}-\mathcal{P}$ , where $\mathcal{P}$ is the operator on $L<^>2(\pi)$ that maps $f\mapsto\int f(y)P(\cdot,\mathrm{d}y)$ and similarly for $\mathcal{Q}$ , is positive on $L<^>2(\pi)$ , i.e. $\langle f,\left(\mathcal{Q}-\mathcal{P}\right)f\rangle\geq0$ for every $f\in L<^>2(\pi)$ (previous proofs for general state spaces use technical results from monotone operator function theory). We use this result to show that under mild conditions, sandwich variants of data augmentation algorithms efficiency-dominate the original algorithm. We also provide other easy-to-check sufficient conditions for efficiency dominance, some of which are generalized from the finite state space case. We also provide a proof based on that of Tierney (1998, Ann. Appl. Prob. 8, 1-9) that Peskun dominance is a sufficient condition for efficiency dominance for reversible kernels. Using these results, we show that Markov kernels formed by random selection of other 'component' Markov kernels will always efficiency-dominate another Markov kernel formed in this way, as long as the component kernels of the former efficiency-dominate those of the latter. These results on the efficiency dominance of combining component kernels generalizes the results on the efficiency dominance of combined chains introduced by Neal and Rosenthal (2024, J. Appl. Prob. 62, 188-208) from finite state spaces to general state spaces.
In this paper, we solve an exit probability game between two players, each of whom controls a linear diffusion process. One player controls its process to minimize the probability that the difference of the processes reaches a low level before it reaches a high level, while the other player aims to maximize the probability. By solving the Bellman-Isaacs equations, we find the sub-value and sup-value functions of the game in explicit forms, which are twice continuously differentiable. The optimal plays associated with the sub-value and sup-value are also found explicitly.
We consider the random series-parallel graph introduced by Hambly and Jordan (2004 Adv. Appl. Probab. 36, 824-838), which is a hierarchical graph with a parameter p is an element of [0, 1]. The graph is built recursively: at each step, every edge in the graph is either replaced with probability p by a series of two edges, or with probability 1 -p by two parallel edges, and the replacements are independent of each other and of everything up to then. At the nth step of the recursive procedure, the distance between the extremal points on the graph is denoted by D-n(p). It is known that D-n(p) possesses a phase transition at p = pc := 1/2; more precisely, 1/nlogE[D-n(p) -> alpha(p) when n ->infinity, with alpha(p) > 0 for p > Pc and alpha(p) = 0 for p <= Pc. We study the exponent alpha(p) in the slightly supercritical regime p=p(c) +epsilon. Our main result says that as epsilon -> 0(+), alpha(pc + epsilon) behaves like root zeta(2)epsilon where zeta(2):= pi(2)/6.
In this paper, we investigate a class of McKean-Vlasov stochastic differential equations (SDEs) with L & eacute;vy-type perturbations. We first establish the existence and uniqueness theorem for the solutions of the McKean-Vlasov SDEs by utilizing an Eulerlike approximation. Then, under suitable conditions, we demonstrate that the solutions of the McKean-Vlasov SDEs can be approximated by the solutions of the associated averaged McKean-Vlasov SDEs in the sense of mean square convergence. In contrast to existing work, a novel feature of this study is the use of a much weaker condition, locally Lipschitz continuity in the state variables, allowing for possibly superlinearly growing drift, while maintaining linearly growing diffusion and jump coefficients. Therefore, our results apply to a broader class of McKean-Vlasov SDEs.
We consider an optimal stopping problem of a linear diffusion under Poisson constraint where the agent can adjust the arrival rate of new stopping opportunities. We assume that the agent may switch the rate of the Poisson process between two values. Maintaining the lower rate incurs no cost, whereas the higher rate requires effort that is captured by a cost function c. We study a broad class of payoff functions, cost functions and diffusion dynamics, for which we explicitly characterize the solution to the constrained stopping problem. We also characterize the case where switching to the higher rate is always suboptimal. The results are illustrated with two examples.
We study a two-dimensional discounted optimal stopping zero-sum (or Dynkin) game related to perpetual redeemable convertible bonds expressed as game (or Israeli) options in a model of financial markets in which the behaviour of the ex-dividend price of a dividend-paying asset follows a generalized geometric Brownian motion. It is assumed that the dynamics of the random dividend rate of the asset paid to shareholders are described by the mean-reverting filtering estimate of an unobservable continuous-time Markov chain with two states. It is shown that the optimal exercise (conversion) and withdrawal (redemption) times forming a Nash equilibrium are the first times at which the asset price hits either lower or upper stochastic boundaries being monotone functions of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundaries are regular for the stopping region relative to the resulting two-dimensional diffusion process and that the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the optimal stopping boundaries are determined as a unique solution to the associated coupled system of nonlinear Fredholm integral equations among the couples of continuous functions of bounded variation satisfying certain conditions. We also give a closed-form solution to the appropriate optimal stopping zero-sum game in the corresponding model with an observable continuous-time Markov chain.
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.
For a spectrally negative Lévy process X , consider $g_t$ and its infinitesimal generator. Moreover, with $t\geq 0$ , the last time X is below the level zero before time $\{(g_t,t, X_t), t\geq 0 \}$ the length of a current positive excursion, we derive a general formula that allows us to calculate a functional of the whole path of $U_t\,:\!=\,t-g_t$ . We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process $ (U, X)=\{(U_t, X_t),t\geq 0\}$ in terms of the positive and negative excursions of the process X . As a corollary, we find the joint Laplace transform of $(U_{\mathbf{e}_q}, X_{\mathbf{e}_q})$ , where $\mathbf{e}_q$ is an independent exponential time, and the q -potential measure of the process ( U , X ). Furthermore, using the results mentioned above, we find a solution to a general optimal stopping problem depending on ( U , X ) with an application in corporate bankruptcy. Lastly, we establish a link between the optimal prediction of $g_{\infty}$ and optimal stopping problems in terms of ( U , X ) as per Baurdoux, E. J. and Pedraza, J. M., $L_p$ optimal prediction of the last zero of a spectrally negative Lévy process, Annals of Applied Probability , 34 (2024), 1350–1402.
In this paper, we introduce a non-homogeneous version of the generalized counting process (GCP). We time-change this process by an independent inverse stable subordinator and derive the system of governing differential–integral equations for the marginal distributions of its increments. We then consider the GCP time-changed by a multistable subordinator and obtain its Lévy measure and the distribution of its first passage times. We discuss an application of a time-changed GCP, namely the time-changed generalized counting process-I (TCGCP-I) in ruin theory. A fractional version of the TCGCP-I is studied, and its long-range dependence property is established.
We establish thresholds for the feasibility of random multi-graph alignment in two models. In the Gaussian model, we demonstrate an "all-or-nothing" phenomenon: above a critical threshold, exact alignment is achievable with high probability, while below it, even partial alignment is statistically impossible. In the sparse Erdős-Rényi model, we rigorously identify a threshold below which no meaningful partial alignment is possible and conjecture that above this threshold, partial alignment can be achieved. To prove these results, we develop a general Bayesian estimation framework over metric spaces, which provides insight into a broader class of high-dimensional statistical problems.
