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.
We prove the existence of a stationary IARCH (integrated autoregressive conditionally heteroskedastic) process for a class of models with polynomially decaying coefficients.
Let $\mathrm{P}_K(n)$ be the probability that n points $z_1,\ldots,z_n$ picked uniformly and independently in K, a compact convex polygon in $\mathbb{R}<^>2$ with non-empty interior, are in convex position, that is, form the vertex set of a convex polygon. In this paper, the exact asymptotic behaviour of $\mathrm{P}_K(n)$ is determined as $n o+\infty$ . This improves on a famous result of B & aacute;r & aacute;ny [1999 Ann. Probab. 27, 2020-2034] (yet valid for a general convex set K) and a result initiated in the case where K is a regular convex polygon (Morin [2025 Adv. Appl. Probab. 57, 811-870]).
While modelling deterioration or ageing of devices, first-passage times of Markov processes play a significant role, especially when the devices are subject to shocks and wear during their operation. In view of this, obtaining sufficient conditions for first-passage times to belong to specific ageing families constitutes an important problem. There exists a rich literature dealing with this class of problems; see for example [11], [18], [35], [67]. We address the same problem in the context of some new ageing classes such as DMTTF (IMTTF), IMIT, and DRHR. In this connection, some issues in [11] have also been investigated. We wind up by including certain examples to highlight the practical relevance of the results.
This work studies time averages of an observable $h(t,X_t)$ , where $X_t$ is the solution to a time-inhomogeneous stochastic differential equation (SDE) driven by drift, b(t, x), and diffusion, $\sigma(t{,}{\kern.5pt}x)$ , that change sufficiently slowly in time. In this quasistatic regime we derive an approximation to the time average that is computable from properties of the time-homogeneous SDEs driven by $b(t,\cdot)$ and $\sigma(t,\cdot)$ with fixed t; specifically, we utilize $\log$ -Sobolev inequalities for the instantaneous invariant distribution and generator for each t. We obtain explicit non-asymptotic error bounds on this quasistatic approximation, both in the form of concentration inequalities and bounds on the expected value. The error bounds demonstrate a competition between the speed of convergence to the instantaneous invariant distributions and their rate of change, matching the intuition that underlies the quasistatic approximation.
This work is devoted to the theoretical and numerical derivation of the moving average coefficients for a first-order autoregressive random field $\{X(\mathbf{t}),\, \mathbf{t}\in \mathbb{Z}<^>{d}\}$ where $d\geq 2$ and $\mathbb{Z}<^>d$ is the lattice of points with integer coordinates in the d-dimensional Euclidean space. We develop formulations for the autocorrelation function, the forecast recipe and the forecast error. A sufficient condition for causality is also provided, in addition to an algorithm and corresponding Wolfram Mathematica code for the numerical computation of the moving average coefficients.
Recent investigations have argued that there is a simple explicit representation for the Kolmogorov constant c associated with the subcritical Galton-Watson branching process. We exhibit examples showing that although this representation can be valid, it more often is not. Our work is presented in terms of the limiting conditional mean population size $\mu=c<^>{-1}$ . The analogous quantity for the Markov branching process is denoted by $\widehat\mu$ . We show that the simple representation put forward for $\widehat\mu$ in fact is an upper bound that is attained only if the offspring-number probability-generating function is quadratic. The conditional mean $\mu$ is the limit of a computable increasing sequence $(\mu_n$ ). Estimates of n are determined ensuring that, for any small positive number $\varepsilon$ , $0\lt\mu-\mu_n\le \varepsilon$ .
We study a stochastic control problem where the underlying process follows a spectrally negative L & eacute;vy process. A controller can continuously increase the process but only decrease it at independent Poisson arrival times. We show the optimality of the periodic-classical barrier strategy, which increases the process whenever it would fall below some lower barrier and decreases it whenever it is observed above a higher barrier. An optimal strategy and the value function are written semi-explicitly using scale functions. Numerical results are also given.
A stochastic model for the spread of an SIR (susceptible $\to$ infective $\to$ removed) epidemic is considered. Infectives have independent and identically distributed infectivity profiles, which describe their infectiousness as a function of time since infection. The individual-to-individual infection rate depends also on the number of susceptibles present in the population. Exact results are derived for the distribution of statistics defined on the final outcome of the epidemic, including its final size. These are proved by using a generalisation of a Sellke construction to show that the distribution of the final outcome of the epidemic is the same as that of an associated discrete-time epidemic process, in which infectives are considered one at a time, and exploiting connection with death processes to analyse the final outcome of the latter. The results generalise easily to multipopulation epidemics.
Fractional Brownian motion, with its long-time correlated increments, has been applied in many fields in recent years. Since volatility was shown to be rough by Gatheral, Jaisson, and Rosenbaum, fractional Brownian motion has gained popularity as a financial model. In this work, we revisit the definitions and properties of the univariate and multivariate fractional Brownian motions, and consider four simulation methods. We demonstrate the issues associated with applying the standard Euler scheme for simulating stochastic processes driven by fractional Brownian motion with H < 1/2 (which we call the rough models). We then introduce a novel approximate method for simulating such rough models based on the fast algorithm by Ma and Wu, which accounts for a factor of 10 speedup. Finally, we consider applications of these methods to option pricing.
Let X-1,...,X-n be independent integers distributed uniformly on [M], M >= 2. A partition S of [n] into nu non-empty subsets S-1,...,S-nu is called perfect if all nu values Sigma X-j is an element of S alpha (j) are equal. For a perfect partition to exist, Sigma(j) X-j has to be divisible by nu . In 2001, for nu=2 , Christian Borgs, Jennifer Chayes, and the author proved that, conditioned on Sigma(j) X-j being even, with high probability a perfect partition exists if kappa:= lim n/log M>1/log 2 , and that with high probability no perfect partition exists if kappa<1/log 2 . Responding to a question by George Varghese, we prove that for nu >= 3 with high probability no perfect partition exists if kappa<2/log nu , which is twice as large as the naive threshold 1/log 3 for nu=3 . We identify the range of kappa where the expected number of perfect partitions is exponentially high. We show that for kappa> 2(nu-1)/log[(1-2 nu(-2))(-1)] the total number of perfect partitions is exponentially high with probability greater than or similar to(1+nu(2))(-1),i..e. below 1/nu , the limiting probability that Sigma(j) X-j is divisible by nu .
This paper introduces a novel expectation-maximization (EM) algorithm for estimating general phase-type (PH) distributions from left-truncated and right-censored (LTRC) data, a common challenge in survival analysis. The proposed algorithm is highly efficient with computational complexity that scales with the number of nonzero elements in the generator matrix. This feature makes the estimation of high-dimensional, sparse PH models computationally tractable and enables the practical use of the computationally intensive extended information criterion for model selection. Numerical experiments demonstrate its significant speed advantage over a modern benchmark and the applicability of PH models to complex lifetime data.
We investigate the limiting spectral distribution of a noncentral unified matrix model defined by $\boldsymbol{\Omega}(\mathbf{X}) = ({(\mathbf{X}\mathbf{P}_1+\mathbf{A})(\mathbf{X}\mathbf{P}_1+\mathbf{A})'}/{n_1}) ({\mathbf{X}\mathbf{P}_2\mathbf{X}'}/{n_2})<^>{-1}$ , where $\mathbf{X}=(X_{ij})_{p\times n}$ is a random matrix with independent and identically distributed real entries having zero mean and finite second moment. $\mathbf{A}$ is a $p\times n$ nonrandom matrix. The matrices $\mathbf{P}_1$ and $\mathbf{P}_2$ are projection matrices satisfying $\mathrm{rank}(\mathbf{P}_1)=n_1$ , $\mathrm{rank}(\mathbf{P}_2)=n_2$ , and $\mathbf{P}_1\mathbf{P}_2=0$ . When $\mathbf{P}_1$ and $\mathbf{P}_2$ are random, they are assumed to be independent of $\mathbf{X}$ . When $p/n_1\to c_1\in(0,\infty)$ and $p/n_2\to c_2\in(0,1)$ , we establish the almost sure convergence of the empirical spectral distribution of $\boldsymbol{\Omega}$ to a deterministic limiting distribution. Furthermore, we show that this limiting distribution coincides with that of the noncentral F-matrix, thus revealing a deep connection between the proposed model and classical multivariate analysis.
We consider the asymptotic behaviour of the expectation of the perimeter deviation of a uniform random spherical disc-polygon in a spherical spindle convex disc with smooth boundary. We also introduce the notion of duality on the sphere, define a model of random circumscribed disc-polygons, and determine some asymptotic results about them.
Consider a random walk in a time-inhomogeneous random environment. When the environment is stationary and ergodic, we identify a quenched harmonic function for almost every realization of the environment. This function allows us to define a random walk in a random environment conditioned to stay positive, using Doob's h-transform.
We consider general discrete-time multitype branching processes on a countable set X. According to these processes, a particle of type $x\in X$ generates a random number of children and chooses their type in X, not necessarily independently nor with the same law for different parent types. We introduce a new type of stochastic ordering of multitype branching processes, generalising the germ order introduced by Hutchcroft, which relies on the generating function of the process. We prove that given two multitype branching processes with laws ${\boldsymbol{\mu}}$ and ${\boldsymbol{\nu}}$ respectively, with ${\boldsymbol{\mu}}\ge{\boldsymbol{\nu}}$ , then in every set where there is survival according to ${\boldsymbol{\nu}}$ , there is also survival according to ${\boldsymbol{\mu}}$ . Moreover, in every set where there is strong survival according to ${\boldsymbol{\nu}}$ , there is also strong survival according to ${\boldsymbol{\mu}}$ , provided that the supremum of the global extinction probabilities for the ${\boldsymbol{\nu}}$ process, taken over all starting points x, is strictly smaller than 1. New conditions for survival and strong survival for inhomogeneous multitype branching processes are provided. We also extend a result of Moyal which claims that, under some conditions, the global extinction probability for a multitype branching process is the only fixed point of its generating function, whose supremum over all starting coordinates may be smaller than 1.
We consider a generalization of the forest fire model on $\mathbb{Z}_+$ with ignition at zero only, studied by Volkov (2009 ALEA 6, 399-414). Unlike that model, we allow delays in the spread of the fires and the non-zero burning time of individual 'trees'. We obtain some general properties for this model, which cover, among others, the phenomenon of an 'infinite fire', not present in the original model.
We investigate some investment problems related to maximizing the expected utility of the terminal wealth in a continuous-time It & ocirc;-Markov additive market. In this market, the prices of financial assets are described by Markov additive processes that combine L & eacute;vy processes with regime-switching models. We give explicit expressions for the solutions to the portfolio selection problem for the hyperbolic absolute risk aversion (HARA) utility, the exponential utility, and the extended logarithmic utility. In addition, we demonstrate that the solutions for the HARA utility are stable in terms of weak convergence when the parameters vary in a suitable way.
