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Explicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some
-algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the
-algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.
In this paper, we show that the numbers of t-stack sortable n-permutations with k − 1 descents satisfy central and local limit theorems for t = 1, 2, n − 1 and n − 2. This result, in particular, gives an affirmative answer to Shapiro's question about the asymptotic normality of the Narayana numbers.
For independent exponentially distributed random variables
, with distinct rates
we consider sums
which follow generalized exponential mixture distributions. We provide novel explicit results on the conditional distribution of the total sum
given that a subset sum
exceeds a certain threshold value
, and vice versa. Moreover, we investigate the characteristic tail behavior of these conditional distributions for
. Finally, we illustrate how our probabilistic results can be applied in practice by providing examples from both reliability theory and risk management.
We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.
In this paper, we introduce a new large family of Lévy-driven point processes with (and without) contagion, by generalising the classical self-exciting Hawkes process and doubly stochastic Poisson processes with non-Gaussian Lévy-driven Ornstein–Uhlenbeck-type intensities. The resulting framework may possess many desirable features such as skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the ‘contagion’ or feedback effects, which could be very useful for modelling event arrivals in finance, economics, insurance, and many other fields. We characterise the distributional properties of this new class of point processes and develop an efficient sampling method for generating sample paths exactly. Our simulation scheme is mainly based on the distributional decomposition of the point process and its intensity process. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of our scheme. Moreover, we use portfolio risk management as an example to show the applicability and flexibility of our algorithms.
We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.
We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.
A family of generalized ageing intensity functions of univariate absolutely continuous lifetime random variables is introduced and studied. They allow the analysis and measurement of the ageing tendency from various points of view. Some of these generalized ageing intensities characterize families of distributions dependent on a single parameter, while others determine distributions uniquely. In particular, it is shown that the elasticity functions of various transformations of distributions that appear in lifetime analysis and reliability theory uniquely characterize the parent distribution. Moreover, the recognition of the shape of a properly chosen generalized ageing intensity estimate admits a simple identification of the data lifetime distribution.
The steepest increase property of phase-type (PH) distributions was first proposed in O’Cinneide (1999) and proved in O’Cinneide (1999) and Yao (2002), but since then has received little attention in the research community. In this work we demonstrate that the steepest increase property can be applied for proving previously unknown moment bounds of PH distributions with infinite or finite support. Of special interest are moment bounds free of specific PH representations except the size of the representation. For PH distributions with infinite support, it is shown that such a PH distribution is stochastically smaller than or equal to an Erlang distribution of the same size. For PH distributions with finite support, a class of distributions which was introduced and investigated in Ramaswami and Viswanath (2014), it is shown that the squared coefficient of variation of a PH distribution with finite support is greater than or equal to 1/(m(m + 2)), where m is the size of its PH representation.
We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry–Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimension d=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimension d≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.
In this paper, we discuss new bounds and approximations for tail probabilities of certain discrete distributions. Several different methods are used to obtain bounds and/or approximations. Excellent upper and lower bounds are obtained for the Poisson distribution. Excellent approximations (and not bounds necessarily) are also obtained for other discrete distributions. Numerical comparisons made to previously proposed methods demonstrate that the new bounds and/or approximations compare very favorably. Some conjectures are made.
We use the Stein‒Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. We treat the case that the fixed graph is a simple graph and that it has multiple edges. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman (2011).
It is well known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also state explicit Poisson approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincaré inequality and in a normal approximation result.
We study the tail asymptotic of subexponential probability densities on the real line. Namely, we show that the n-fold convolution of a subexponential probability density on the real line is asymptotically equivalent to this density multiplied by n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular subexponential functions and use it to find an analogue of Kesten's bound for functions on ℝd. The results are applied to the study of the fundamental solution to a nonlocal heat equation.
Poisson-like behavior for event count data is ubiquitous in nature. At the same time, differencing of such counts arises in the course of data processing in a variety of areas of application. As a result, the Skellam distribution – defined as the distribution of the difference of two independent Poisson random variables – is a natural candidate for approximating the difference of Poisson-like event counts. However, in many contexts strict independence, whether between counts or among events within counts, is not a tenable assumption. Here we characterize the accuracy in approximating the difference of Poisson-like counts by a Skellam random variable. Our results fully generalize existing, more limited, results in this direction and, at the same time, our derivations are significantly more concise and elegant. We illustrate the potential impact of these results in the context of problems from network analysis and image processing, where various forms of weak dependence can be expected.
Let P be the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vector πT, and let (n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n + 1) x (n + 1) northwest corner truncation of P arbitrarily, with invariant probability vector (n)πT. We derive computable V-norm bounds on the error between πT and (n)πT in terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 as n tends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.
By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.
For all α > 0 and real random variables X, we establish sharp bounds for the smallest and the largest deviation of αX from the logarithmic distribution also known as Benford's law. In the case of uniform X, the value of the smallest possible deviation is determined explicitly. Our elementary calculation puts into perspective the recurring claims that a random variable conforms to Benford's law, at least approximately, whenever it has large spread.
We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.