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The usual stochastic order and the likelihood ratio order between probability distributions on the real line are reviewed in full generality. In addition, for the distribution of a random pair (X, Y), it is shown that the conditional distributions of Y, given X = x, are increasing in x with respect to the likelihood ratio order if and only if the joint distribution of (X, Y) is totally positive of order two (TP2) in a certain sense. It is also shown that these three types of constraints are stable under weak convergence, and that weak convergence of TP2 distributions implies convergence of the conditional distributions just mentioned. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean-Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem. [ABSTRACT FROM AUTHOR]

Parallel tempering (PT) methods are a popular class of Markov chain Monte Carlo schemes used to sample complex high‐dimensional probability distributions. They rely on a collection of N interacting auxiliary chains targeting tempered versions of the target distribution to improve the exploration of the state space. We provide here a new perspective on these highly parallel algorithms and their tuning by identifying and formalizing a sharp divide in the behaviour and performance of reversible versus non‐reversible PT schemes. We show theoretically and empirically that a class of non‐reversible PT methods dominates its reversible counterparts and identify distinct scaling limits for the non‐reversible and reversible schemes, the former being a piecewise‐deterministic Markov process and the latter a diffusion. These results are exploited to identify the optimal annealing schedule for non‐reversible PT and to develop an iterative scheme approximating this schedule. We provide a wide range of numerical examples supporting our theoretical and methodological contributions. The proposed methodology is applicable to sample from a distribution π with a density L with respect to a reference distribution π0 and compute the normalizing constant ∫Ldπ0. A typical use case is when π0 is a prior distribution, L a likelihood function and π the corresponding posterior distribution. [ABSTRACT FROM AUTHOR]

Exact and asymptotic distributional properties are discussed in detail for two mean-squared error consistent point predictors of future-generalized order statistics (GOSs) based on two-parameter exponential distribution. These predictors work even if some observed data were missing. For each point predictor, the asymptotic distribution of the normalized difference between the future GOS and its point predictor is derived, when the scale parameter is known or unknown. It is revealed that the asymptotic distributions of these normalized differences are equal when the scale parameter is known. Two asymptotic prediction intervals of the future GOS are constructed whenever the scale parameter is known or unknown. Furthermore, two tests of outliers are proposed relying on the point predictors. Finally, a simulation study is conducted and a real data set is analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

We study the limit distribution of partial sums with a random number of terms following a class of mixed Poisson distributions. The resulting weak limit is a mixture between a normal distribution and an exponential family, which we call by normal exponential family (NEF) laws. A new stability concept is introduced and a relationship between α -stable distributions and NEF laws is established. We propose the estimation of the NEF model parameters through the method of moments and also by the maximum likelihood method via an Expectation–Maximization algorithm. Monte Carlo simulation studies are addressed to check the performance of the proposed estimators, and an empirical illustration of the financial market is presented. [ABSTRACT FROM AUTHOR]

In this paper, extending the ideas given by (Deheuvels 1979) for the empirical copula, we first assume that we are sampling from the product copula, and we construct C m n the d-sample copula of order m, which is an estimator of C (m) the checkerboard approximation of order m, see (Li et al. 1997), then we study the distribution of the number of observations in each of the boxes generated by the regular partition of order m in [ 0 , 1 ] 2 , and we also give the corresponding moments and correlations. We also prove the weak convergence of the sample process m n ( C m m n − C (m) ) to a centered Gaussian process. In fact, we use the known results of the weak convergence for the empirical process, to prove the weak convergence of the sample process. We first show that the sample copula can be written as a linear functional of the empirical copula, and then we observe that this functional is Hadamard differentiable. Therefore, we can apply the delta method to get the weak convergence of the sample process. Finally, we performed several simulations of the sample process at a given point of [ 0 , 1 ] 2 , to study the sample size needed to get the convergence to a centered Gaussian process. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit distribution functions of the (lower-lower), (upper-upper) and (lower-upper) extreme and central-central m-generalised order statistics (m–GOS) of stationary Gaussian sequences under an equi-correlated set up, when the random sample size is assumed to converge weakly and independent of the basic variables. Moreover, sufficient conditions for a weak convergence of generalised quasi-range with random indices are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the limit distributions of upper and lower record values of a stationary Gaussian sequence under an equi-correlated set up. Moreover, the class of limit distribution functions (df's) of the joint upper (and the lower) record values of a stationary Gaussian sequence is fully characterized. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-mid-range, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit df's of these statistics are derived. [ABSTRACT FROM AUTHOR]

Sufficient conditions for the weak convergence of the distributions of the random variables (1 - x)ξx as x → 1- to the limiting gamma-distribution are put forward. The random variable ξx has power-series distribution with radius of convergence 1 and parameter x ∈ (0, 1). Limit theorems for the probabilities P{ξx = k} are proposed. Asymptotic expansions of local probabilities are derived for sums of independent identically distributed variables with the same distribution as ξx in a triangular array with x → 1-. For the corresponding general allocation scheme, local limit theorems for the joint distributions of the occupancies of the cells are obtained. [ABSTRACT FROM AUTHOR]

A common assumption in pair-copula constructions is that the copula of the conditional distribution of two random variables given a covariate does not depend on the value of that covariate. Two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. On the one hand, the best possible rates for estimating the marginal conditional distribution functions are slower than the parametric one. On the other hand, the invariance of the conditional copula given the value of the covariate suggests the possibility of parametric convergence rates. The more optimistic intuition is shown to be correct, confirming a conjecture supported by extensive Monte Carlo simulations by Hobæk Haff and Segers (2015) and improving upon the nonparametric rate obtained theoretically by Gijbels et al. (2015). The novelty of the proposed approach lies in a double smoothing procedure for the estimator of the marginal conditional distribution functions. The copula estimator itself is asymptotically equivalent to an oracle empirical copula, as if the marginal conditional distribution functions were known. [ABSTRACT FROM AUTHOR]

We provide conditions on dependent and on non-stationary random variables X n ensuring that the mantissa of the sequence of products ∏ 1 n X k is almost surely distributed following Benford’s law or converges in distribution to Benford’s law. This is achieved through proving new generalizations of Lévy’s and Robbins’s results on distribution modulo 1 of sums of independent random variables. [ABSTRACT FROM AUTHOR]

We consider the limit distribution of a sum of strongly additive arithmetic functions with arguments runningthrough shifted primes. We obtain sufficient and necessary conditions for the weak convergence of distributions of such sums to the discrete uniform law. We study the case where the functions take values 0 or 1 on primes. [ABSTRACT FROM AUTHOR]

We study a class of non-stationary shot noise processes which have a general arrival process of noises with non-stationary arrival rate and a general shot shape function. Given the arrival times, the shot noises are conditionally independent and each shot noise has a general (multivariate) cumulative distribution function (c.d.f.) depending on its arrival time. We prove a functional weak law of large numbers and a functional central limit theorem for this new class of non-stationary shot noise processes in an asymptotic regime with a high intensity of shot noises, under some mild regularity conditions on the shot shape function and the conditional (multivariate) c.d.f. We discuss the applications to a simple multiplicative model (which includes a class of non-stationary compound processes and applies to insurance risk theory and physics) and the queueing and work-input processes in an associated non-stationary infinite-server queueing system. To prove the weak convergence, we show new maximal inequalities and a new criterion of existence of a stochastic process in the space D given its consistent finite dimensional distributions, which involve a finite set function with the superadditive property. [ABSTRACT FROM AUTHOR]

In this short note we consider the finite-dimensional distributions of sets of states generated by dispersing billiards with a random initial condition. We establish a functional correlation bound on the distance between the finite-dimensional distributions and corresponding product distributions. We demonstrate the usefulness of the bound by showing that it implies several limit theorems. The purpose of this note is to provide a tool facilitating the study of more general functionals of the billiard process. [ABSTRACT FROM AUTHOR]

Given an Itō semimartingale with a time-homogeneous jump part observed at high frequency, we prove weak convergence of a normalized truncated empirical distribution function of the Lévy measure to a Gaussian process. In contrast to competing procedures, our estimator works for processes with a non-vanishing diffusion component and under simple assumptions on the jump process. [ABSTRACT FROM AUTHOR]

In this paper, we show that for t > 0, the joint distribution of the past {Wt-s : 0 ≤ s ≤ t} and the future {Wt+s : s ≥ 0} of a d-dimensional standard Brownian motion (Ws), conditioned on {Wt ∈ U}, where U is a bounded open set in Rd, converges weakly in C[0, ∞) × C[0, ∞) as t → ∞. The limiting distribution is that of a pair of coupled processes Y + B¹, Y + B² where Y, B¹, B² are independent, Y is uniformly distributed on U and B¹, B² are standard d-dimensional Brownian motions. Let σt, dt be respectively, the last entrance time before time t into the set U and the first exit time after t from U. When the boundary of U is regular, we use the continuous mapping theorem to show that the limiting distribution as t → ∞ of the four dimensional vector with components (Wσt, t - σt, Wdt, dt - t), conditioned on {Wt ∈ U}, is the same as that of the four dimensional vector whose components are the place and time of first exit from U of the processes Y + B¹ and Y + B² respectively. [ABSTRACT FROM AUTHOR]

Directed information or its variants are utilized extensively in the characterization of the capacity of channels with memory and feedback, nonanticipative lossy data compression, and their generalizations to networks. In this paper, we derive several functional and topological properties of directed information, defined on general abstract alphabets (complete separable metric spaces), using the topology of weak convergence of probability measures. These include the convexity of the set of consistent distributions, which uniquely define causally conditioned distributions, convexity, and concavity of directed information with respect to the sets of consistent distributions, weak compactness of such sets of distributions, their joint distributions, and their marginals. Furthermore, we show lower semicontinuity of directed information, and under certain conditions, we also establish continuity. Finally, we derive variational equalities for directed information, including sequential versions. These may be viewed as the analog of the variational equalities of mutual information (utilized in Blahut–Arimoto algorithms). In summary, we extend the basic functional and topological properties of mutual information to directed information. These properties are discussed throughout this paper, in the context of extremum problems of directed information. [ABSTRACT FROM PUBLISHER]

For each n ≥ 1, let $\{ X_{in}, \quad i \geqslant 1 \}$ be independent copies of a nonnegative continuous stochastic process X = ( X( s)) indexed by a compact metric space S. We are interested in the process of partial maxima $\tilde M_{n}(t,s) =\max \{ X_{in}(s), 1 \leqslant i\leqslant [nt] \},\quad t\geq 0,\ s\in S,$ where the brackets [ ⋅ ] denote the integer part. Under a regular variation condition on the sequence of processes X, we prove that the partial maxima process $\tilde M_{n}$ weakly converges to a superextremal process $\tilde M$ as $n\to \infty $. We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko. [ABSTRACT FROM AUTHOR]

The main result of this contribution is the derivation of the exact asymptotic behavior of the supremum of a class of α(t)-locally stationary Gaussian random fields. We present two applications of our result: the first one deals with the extremes of aggregate multifractional Brownian motions, whereas the second one establishes the exact asymptotics of the supremum of the Χ-process generated by multifractional Brownian motions. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions for the weak convergence of the generalized range, midrange, extremal quotient, and extremal product are obtained. The classes of possible non degenerate limit distribution functions of these simple statistics are characterized. Comparison study between these statistics with some examples for the most important distribution functions are given. [ABSTRACT FROM PUBLISHER]

To avoid finding the stationary distributions of stochastic differential equations by solving the nontrivial Kolmogorov–Fokker–Planck equations, the numerical stationary distributions are used as the approximations instead. This paper is devoted to approximate the stationary distribution of the underlying equation by the Backward Euler–Maruyama method. Currently existing results (Mao et al., 2005; Yuan et al., 2005; Yuan et al., 2004) are extended in this paper to cover larger range of nonlinear SDEs when the linear growth condition on the drift coefficient is violated. [ABSTRACT FROM AUTHOR]

We prove two functional limit theorems for empirical multiparameter second moment functions (generalizing Ripley's K-function) obtained from a homogeneous Poisson point field observed in an unboundedly expanding convex sampling window W in ℝ. The cases of known and unknown (estimated) intensity lead to distinct Gaussian limits and require quite different proofs. Further we determine the limit distributions of the maximal deviation and the integrated squared distance between empirical and true multiparameter second moment function. These results give rise to construct goodness-of-fit tests for checking the hypothesis that a given point pattern is completely spatially random (CSR), that is, a realization of a homogeneous Poisson process. [ABSTRACT FROM AUTHOR]

The class of limit distribution functions (df's) of multivariate order statistics from independent and identical random vectors with random sample size is fully characterized. Two cases in this study are considered, the first case is when the random sample size is assumed to be independent of all the basic random vectors, and the second case is when the interrelation of the random size and the basic random vectors is not restricted. [ABSTRACT FROM AUTHOR]

Let { X( t), t ≥ 0} be a standard (zero-mean, unit-variance) stationary Gaussian process with correlation function r(·) and continuous sample paths. In this paper, we consider the maxima M( T) = max{ X( t), ∀ t ∈ [0, T]} with random index T, where T/ T converges to a non-degenerate distribution or to a positive random variable in probability, and show that the limit distribution of M( T) exists under some additional conditions related to the correlation function r(·). [ABSTRACT FROM AUTHOR]

We consider a broad class of fair leader election algorithms, and study the duration of contestants (the number of rounds a randomly selected contestant stays in the competition) and the overall cost of the algorithm. We give sufficient conditions for the duration to have a geometric limit distribution (a perpetuity built from Bernoulli random variables), and for the limiting distribution of the total cost (after suitable normalization) to be a perpetuity. For the duration, the proof is established via convergence (to 0) of the first-order Wasserstein distance from the geometric limit. For the normalized overall cost, the method of proof is also convergence of the first-order Wasserstein distance, augmented with an argument based on a contraction mapping in the first-order Wasserstein metric space to show that the limit approaches a unique fixed-point solution of a perpetuity distributional equation. The use of these two steps is commonly referred to as the contraction method. [ABSTRACT FROM AUTHOR]

The recently introduced two-parameter infinitely-many-neutral-alleles model extends the celebrated one-parameter version (which is related to Kingman's distribution) to diffusive two-parameter Poisson-Dirichlet frequencies. In this paper we investigate the dynamics driving the species heterogeneity underlying the two-parameter model. First we show that a suitable normalization of the number of species is driven by a critical continuous-state branching process with immigration. Secondly, we provide a finite-dimensional construction of the two-parameter model, obtained by means of a sequence of Feller diffusions of Wright-Fisher flavor which feature finitely many types and inhomogeneous mutation rates. Both results provide insight into the mathematical properties and biological interpretation of the two-parameter model, showing that it is structurally different from the one-parameter case in that the frequency dynamics are driven by state-dependent rather than constant quantities. [ABSTRACT FROM AUTHOR]

We obtain the convergence rate for the distribution of the fluctuation of almost critical branching processes with immigration to the normal law. [ABSTRACT FROM AUTHOR]

In this work, we consider the nonparametric estimation of quality adjusted lifetime (QAL) distribution in a simple illness-death model. We first derive the expression of QAL distribution in terms of the distribution of sojourn time in each health state. Next we substitute the estimate of sojourn time distributions in the expression of QAL distribution to obtain its estimate. Consistency and asymptotic normality of the proposed nonparametric estimator are established. Estimation in the presence of some missing data on the transition time to illness is also discussed. We conduct a simulation study to investigate the performance of the proposed estimator. For illustration, we analyse a data set of the Stanford Heart Transplant program. Extension to multistate progressive model is discussed along with an analysis of International Breast Cancer Study Group (IBCSG) Trial V data. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak convergence of the random maximum of independent and non-identical random vectors. When the random sample size is assumed to be independent of the basic variables and its distribution function is assumed to converge weakly to a non-degenerate limit, the necessary and sufficient conditions for the weak convergence of the random maximum are derived. An illustrative example is given. [ABSTRACT FROM AUTHOR]

In this study, asymptotic expansions of the moments of the maximum ( M(β)) of Gaussian random walk with negative drift ( − β), β > 0, are established by using Bell Polynomials. In addition, the weak convergence theorem for the distribution of the random variable Y(β) ≡ 2 β M(β) is proved, and the explicit form of the limit distribution is derived. Moreover, the approximation formulas for the first four moments of the maximum of a Gaussian random walk are obtained for the parameter β ∈ (0.5, 3.2] using meta-modeling. [ABSTRACT FROM AUTHOR]

In the paper, a multiplicative version of the distribution of the vector (|ζ(s)|, ζ(s)), where ζ(s) is the Riemann zeta-function, is considered and probabilistic limit theorems are obtained. The dependence between |ζ(s)| and ζ(s) in terms of m-characteristic transforms is also discussed. [ABSTRACT FROM AUTHOR]

Abstract: The peakedness of a random variable (RV) about a point is defined by . A RV is said to be less peaked about than a RV about , denoted by , if for all , i.e., is stochastically larger than . These generalize the original definitions of Birnbaum (1948) who considered the cases where and were symmetric about and , respectively. Statistical inferences about the distribution functions of continuous and under peakedness ordering in the symmetric case have been treated in the literature. Rojo et al. (2007) provided estimators of the distributions in the general case and analyzed their properties. We show that these estimators could have poor asymptotic properties relative to those of the empiricals. We provide improved estimators of the DFs, show that they are consistent, derive the weak convergence of the estimators, compare them with the empirical estimators, and provide formulas for statistical inferences. An example is also used to illustrate our theoretical results. [Copyright &y& Elsevier]

Abstract: The class of limit distribution functions of the random record model is fully characterized. Necessary and sufficient conditions as well as the domains of attraction of the limit distribution functions are obtained. [Copyright &y& Elsevier]

Abstract: Assuming that is a sequence of càdlàg processes converging in distribution to in the Skorohod topology, conditions are given under which converges weakly to in the space , where is a sequence of “smooth” functions converging to . Integrals of this form arise as the objective function for inference about a parameter in a stochastic model. Convergence of these integrals play a key role in describing the asymptotics of the estimator of which optimizes the objective function. We illustrate this with a moving average process. [Copyright &y& Elsevier]

Abstract: For the one-dimensional telegraph process, we obtain explicitly the distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of subnormal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals. [Copyright &y& Elsevier]

We prove an invariance principle under self-normalization by blocks for linear processes with summable filters and i.i.d. innovations in the domain of attraction of the normal distribution. [ABSTRACT FROM AUTHOR]

In survival analysis, the classical Koziol-Green random censorship model is commonly used to describe informative censoring. Hereby, it is assumed that the distribution of the censoring time is a power of the distribution of the survival time. In this article, we extend this model by assuming a general function between these distributions. We determine this function from a relationship between the observable random variables which is described by a copula family that depends on an unknown parameter θ. For this setting, we develop a semi-parametric estimator for the distribution of the survival time in which we propose a pseudo-likelihood estimator for the copula parameter θ. As results, we show first the consistency and asymptotic normality of the estimator for θ. Afterwards, we prove the weak convergence of the process associated to the semi-parametric distribution estimator. Furthermore, we investigate the finite sample performance of these estimators through a simulation study and finally apply it to a practical data set on survival with malignant melanoma. [ABSTRACT FROM AUTHOR]

Abstract: In competing risks studies, the Kaplan–Meier estimators of the distribution functions (DFs) of lifetimes and the corresponding estimators of cumulative incidence functions (CIFs) are used widely when no prior information is available for these distributions. In some cases better estimators of the DFs of lifetimes are available when they obey some inequality constraints, e.g., if two lifetimes are stochastically or uniformly stochastically ordered, or some functional of a DF obeys an inequality in an empirical likelihood estimation procedure. If the restricted estimator of a lifetime differs from the unrestricted one, then the usual estimators of the CIFs will not add up to the lifetime estimator. In this paper we show how to estimate the CIFs in this case. These estimators are shown to be strongly uniformly consistent. In all cases we consider, when the inequality constraints are strict the asymptotic properties of the restricted and the unrestricted estimators are the same, thus providing the asymptotic properties of the restricted estimators essentially “free of charge”. We give an example to illustrate our procedure. [Copyright &y& Elsevier]

Abstract: We construct a family of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener–Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class. [Copyright &y& Elsevier]

In this paper, we prove a limit theorem for the argument of twisted with character automorphic L-functions with an increasing modulus of the character. [ABSTRACT FROM AUTHOR]

In this paper it is proved that the weak convergence on a compact interval to a continuous, strictly increasing limit function, of the distribution function of a record value, implies weak convergence throughout . [ABSTRACT FROM AUTHOR]

Abstract: SiZer (SIgnificant ZERo crossing of the derivatives) is a scale-space visualization tool for statistical inferences. In this paper we introduce a graphical device, which is based on SiZer, for the test of the equality of the mean of two time series. The estimation of the quantile in a confidence interval is theoretically justified by advanced distribution theory. The extension of the proposed method to the comparison of more than two time series is also done using residual analysis. A broad numerical study is conducted to demonstrate the sample performance of the proposed tool. In addition, asymptotic properties of SiZer for the comparison of two time series are investigated. [Copyright &y& Elsevier]

Let X be a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with infinite Lévy measure, let Xϵ be the sum of jumps not exceeding ϵ, and let μ(ϵ) = E[Xϵ(l)]. We study the question of weak convergence of Xϵ/μ(ϵ) as ϵ ↓ 0, in terms of the limit behavior of μ(ϵ)/ϵ. The most interesting case reduces to the weak convergence of Xϵ/ϵ to a subordinator whose marginals are generalized Dickman distributions; we give some necessary and sufficient conditions for this to hold. For a certain significant class of subordinators for which the latter convergence holds, and whose most prominent representative is the gamma process, we give some detailed analysis regarding the convergence quality (in particular, in the context of approximating X itself). This paper completes, in some respects, the study made by Asmussen and Rosiński (2001). [ABSTRACT FROM AUTHOR]

The Marshall and Olkin (1997) parametrization operation is given by [image omitted] where F1 is a given distribution function and α > 0, [image omitted] = 1 - α. In this article, we show that both Fα = Pα(F1) and the generating distribution F1 belong to the same domain of maximal (minimal) (upper record value) (lower record value) attraction. Moreover, it is shown that the weak convergence of any non extreme order statistic (order statistic with variable rank) based on F1, under given normalizing constants, to a non degenerate limit type implies that the same non extreme order statistic based on the family Fα = Pα(F1), α≠1, under the same normalizing constants, does not converge to any non degenerate distribution and vice versa. [ABSTRACT FROM AUTHOR]