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Therefore, the liability cash flow should in what follows be interpreted as either the residual liability cash flow that remains after partial replication or a liability cash flow that is not replicable by financial replication instruments. The value of the liability cash flow is then the sum of the market price of the replicating portfolio and the cost-of-capital margin for the residual liability cash flow (corresponding to the replication error). Keywords: Multi-period valuation; cost-of-capital margin; continuous-time limit; 91G50; 60B12; 91B30 EN Multi-period valuation cost-of-capital margin continuous-time limit 91G50 60B12 91B30 79 106 28 01/27/22 20200701 NES 200701 1 Introduction 1.1 Multi-period cost-of-capital valuation This paper focuses on multi-period cost-of-capital valuation of a cumulative liability cash flow HT ht subject to repeated capital requirements at the beginning of each time period, where the time periods form a partition of HT ht . [Extracted from the article]

The G-Elliptic law under the G-Lindeberg condition for the independent pairs of the entries of a random matrix is proven. [ABSTRACT FROM AUTHOR]

First, weak solutions of generalized stochastic Hamiltonian systems (gsHs) are constructed via essential m-dissipativity of their generators on a suitable core. For a scaled gsHs we prove convergence of the corresponding semigroups and tightness of the weak solutions. This yields convergence in law of the scaled gsHs to a distorted Brownian motion. In particular, the results confirm the convergence of the Langevin dynamics in the overdamped regime to the overdamped Langevin equation. The proofs work for a large class of (singular) interaction potentials including, e.g. potentials of Lennard-Jones type. [ABSTRACT FROM AUTHOR]

Let { X n , n ≥ 1 } be coordinatewise negatively associated random vectors taking values in real separable Hilbert space. In this note we prove the convergence of ∑ n = 1 ∞ n − 1 − α E ( max 1 ≤ k ≤ n | | S k | | − ϵ n α) + and ∑ n = 1 ∞ n − 1 − α log n E ( max 1 ≤ k ≤ n | | S k | | − ϵ n α) + , where S k = X 1 + ⋯ + X k. The present investigation provides the complete moment convergence for the case α r = 1 and generalizes some results in Huan (2015). [ABSTRACT FROM AUTHOR]

For a sequence { X n , n ⩾ 1 } of independent random elements taking values in a Rademacher type p Banach space with the k-th partial sum S k (k ⩾ 1) , we provide necessary and sufficient conditions for the convergence of ∑ n = 1 ∞ 1 n P (max 1 ⩽ k ⩽ n ‖ S k ‖ > ε n α) and ∑ n = 1 ∞ log n n P (max 1 ⩽ k ⩽ n ‖ S k ‖ > ε n α) for every ε > 0 . [ABSTRACT FROM AUTHOR]

In this correspondence, complete convergence theorems are established for weighted row sums { ∑ i = 1 j a n , i V n , i , 1 ≤ j ≤ k n , n ≥ 1 } from arrays { V n , i , 1 ≤ i ≤ k n , n ≥ 1 , k n → ∞ } of random elements taking values in real separable Rademacher type p (1 ≤ p ≤ 2) Banach spaces as well as real separable martingale type p (1 ≤ p ≤ 2) Banach spaces. It is assumed that sup { E ‖ V n , i ‖ p : 1 ≤ i ≤ k n , n ≥ 1 } < ∞. A version of the Rademacher type p complete convergence theorem is also established with random variable weights. Illustrative examples are included. [ABSTRACT FROM AUTHOR]

In this note, we study an infinite reaction network called the stochastic Becker–Döring process, a sub-class of the general coagulation–fragmentation models. We prove pathwise convergence of the process towards the deterministic Becker–Döring equations which improves classical tightness-based results. Also, we show by studying the asymptotic behavior of the stationary distribution, that the phase transition property of the deterministic model is also present in the finite stochastic model. Such results might be interpreted closed to the so-called gelling phenomena in coagulation models. We end with few numerical illustrations that support our results. [ABSTRACT FROM AUTHOR]

We describe the construction of a random walk in a Banach space B with a quasi-orthogonal Schauder basis and show that it is a martingale. Next we prove that under certain additional assumptions the described random walk converges a.s. and in L p B , 1 ≤ p < ∞ , to a random element ξ , which generates a probability measure with support contained in the unit ball B ⊂ B . Moreover, we define the Banach integral with respect to the distribution of ξ for a class of bounded, Borel measurable real-valued functions on B. Next some examples of nonstandard Banach spaces with quasi-orthogonal Schauder bases are presented; furthermore, examples which demonstrate the possibility of applications of all the obtained results in spaces ℓ p , 1 ≤ p < ∞ and L p [ 0 , 1 ] , 1 < p < ∞ are given. [ABSTRACT FROM AUTHOR]

For a double array {Vm,n,m ≥ 1, n ≥ 1} of independent, mean 0 random elements in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space and an increasing double array {bm,n,m ≥ 1, n ≥ 1} of positive constants, the limit law m a x 1 ≤ k ≤ m , 1 ≤ l ≤ n ∥ Σ i = 1 k Σ j = 1 l V i , j ∥ / b m , n → 0 a.c. and in ℒp as m ∨ n → ∞ is shown to hold if Σ m = 1 ∞ Σ n = 1 ∞ E ∥ V m , n ∥ p / b m , n p < ∞ . This strong law of large numbers provides a complete characterization of Rademacher type p Banach spaces. Results of this form are also established when 0 < p ≤ 1 where no independence or mean 0 conditions are placed on the random elements and without any geometric conditions placed on the underlying Banach space. [ABSTRACT FROM AUTHOR]

Given a probability space (Ω,A,P), a complete and separable metric space X with the σ-algebra B of all its Borel subsets and a B⊗A-measurable f:X×Ω→X we consider its iterates fn defined on X×ΩN by f0(x,ω)=x and fn(x,ω)=f(fn-1(x,ω),ωn) for n∈N and provide a simple criterion for the existence of a probability Borel measure π on X such that for every x∈X and for every Lipschitz and bounded ψ:X→R the sequence 1n∑k=0n-1ψ(fk(x,·))n∈N converges in probability to ∫Xψ(y)π(dy). [ABSTRACT FROM AUTHOR]

This paper provides necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces. The main theorem extends Theorem 1.2 in [1] to the double weighted sum setting. The sharpness of the main result is illustrated by showing that the main theorem can fail if we replace the identical distribution condition by a slightly weaker condition, even when the random elements are independent and uniformly almost surely bounded. [ABSTRACT FROM AUTHOR]

We consider various probability distributions {Gn,n≥1} concentrated on the interval [-1,1]⊂R and investigate basic properties of the limit distribution Γ of the Banach random walk in a Banach space B generated by {Gn,n≥1}. In particular, we describe assumptions ensuring that the support of Γ is equal to the unit sphere in B and, on the other hand, we find conditions under which every ball of radius smaller than 1 has a positive measure Γ. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of pairwise and coordinatewise negative dependence for random vectors in Hilbert spaces. Besides giving some classical inequalities, almost sure convergence and complete convergence theorems are established. Some limit theorems are extended to pairwise and coordinatewise negatively dependent random vectors taking values in Hilbert spaces. An illustrative example is also provided. [ABSTRACT FROM AUTHOR]

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37-54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded G/GI/n+GI queue with stationary arrivals, nonexponential service times, n identical servers, and nonexponential patience times. Process-level convergence to non-Markovian Gaussian limits is established as the number of servers goes to infinity for key performance processes such as the waiting times, queue length, abandonment and departure processes. Analytic formulas are developed to characterize the distributions of these Gaussian limits. [ABSTRACT FROM AUTHOR]

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett.69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math.61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep.11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl.119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result. [ABSTRACT FROM AUTHOR]

This paper presents new results on the prediction of linear processes in function spaces. The autoregressive Hilbertian process framework of order one (ARH(1) framework) is adopted. A component-wise estimator of the autocorrelation operator is derived from the moment-based estimation of its diagonal coefficients with respect to the orthogonal eigenvectors of the autocovariance operator, which are assumed to be known. Mean-square convergence to the theoretical autocorrelation operator is proved in the space of Hilbert–Schmidt operators. Consistency then follows in that space. Mean absolute convergence, in the underlying Hilbert space, of the ARH(1) plug-in predictor to the conditional expectation is obtained as well. A simulation study is undertaken to illustrate the large-sample behavior of the formulated component-wise estimator and predictor. Additionally, alternative component-wise (with known and unknown eigenvectors), regularized, wavelet-based penalized, and nonparametric kernel estimators of the autocorrelation operator are compared with the one presented here, in terms of prediction. [ABSTRACT FROM AUTHOR]

We propose an approach to the Baum-Katz theorem for dependent random sequences. Using this result, we obtain the rate of convergence in the strong law of large numbers for sequences of independent random elements in Rademacher type p Banach spaces and for sequences of negatively associated random variables. [ABSTRACT FROM AUTHOR]

This paper is devoted to a study on the structure of tensorial products of periodically correlated autoregressive (PCAR) processes with values in separable Hilbert spaces. It will be demonstrated that the resulting processes are PCAR with values in the space of Hilbert–Schmidt operators. These processes are applied while studying the convergence rate, limiting behavior and asymptotic distribution of the empirical estimators of the covariance operators of PCAR processes. [ABSTRACT FROM PUBLISHER]

We develop a Glivenko–Cantelli theory for monotone, almost additive functions of i.i.d. sequences of random variables indexed by Z d . Under certain conditions on the random sequence, short range correlations are allowed as well. We have an explicit error estimate, consisting of a probabilistic and a geometric part. We apply the results to yield uniform convergence for several quantities arising naturally in statistical physics. [ABSTRACT FROM AUTHOR]

Let $$\{X_k\}_{k \in \mathbb {Z}} \in {\mathbb {L}}^2({\mathcal {T}})$$ be a stationary process with associated lag operators $${\varvec{\mathcal {C}}}_h$$ . Uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues (spectral gap). In addition, the underlying dependence assumptions are optimal in a certain sense, including both short and long memory processes. This allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions. Among other things, convergence to an extreme value distribution is shown. We also discuss how the asymptotic expansions transfer to the long-run covariance operator $${\varvec{\mathcal {G}}}$$ in a general framework. [ABSTRACT FROM AUTHOR]

Conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type p Banach space converges completely to 0 in mean of order p. These conditions for the complete convergence in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. In case the Banach space is not of Rademacher type p, it is proved that the complete convergence in mean of order p of a normed double sum implies a strong law of large numbers. [ABSTRACT FROM AUTHOR]

The authors study the complete moment convergence for weighted sums of arrays of Banach space valued random elements. The results in this paper improve the corresponding theorems of Qiu et al. (Lith Math J 52(3):316-325, 2012). [ABSTRACT FROM AUTHOR]

A two-dimensionally indexed random coefficients autoregressive models (2D−RCAR) and the corresponding statistical inference are important tools for the analysis of spatial lattice data. The study of such models is motivated by their second-order properties that are similar to those of 2D− (G)ARCHwhich play an important role in spatial econometrics. In this article, we study the asymptotic properties of two-stage generalized moment method (2S−GMM) under general asymptotic framework for 2D−RCAmodels. So, the efficiency, strong consistency, the asymptotic normality, and hypothesis tests of 2S−GMMestimation are derived. A simulation experiment is presented to highlight the theoretical results. [ABSTRACT FROM PUBLISHER]

The instant system availability $$S_\tau (t)$$ of a repairable system with the renewal equation was studied. The starting point monotonicity of $$S_\tau (t)$$ was proved and the upper bound of $$S_\tau (t)$$ is also derived. It was found that the interval of instant system availability monotonically decreases. In addition, we provide examples that validate the analytically derived properties of $$S_\tau (t)$$ based on the Lognormal, Gamma and Weibull distributions and the results show that the value of T is slightly smaller than its value defined in Theorem 2. The procedure of using a bathtub as application for this article is also discussed. [ABSTRACT FROM AUTHOR]

Based on iteration of random-valued functions we study the problems of existence, uniqueness and continuous dependence of Lipschitzian solutions $${\varphi}$$ of the equation where P is a probability measure on a $${\sigma}$$ -algebra of subsets of $${\Omega}$$ . [ABSTRACT FROM AUTHOR]

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from Erdős et al. (Adv Math 229(3):1435-1515, 2012; Electron J Probab 18(59):1-58, 2013), by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate (log N)/N and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of Erdős et al. (Adv Math 229(3):1435-1515,2012 ; Electron J Probab 18(59):1-58, 2013), Götze and Tikhomirov (2013). [ABSTRACT FROM AUTHOR]

This note is a continuation of the paper []. Let $${\{X_{n},\ {n\geqq 1}\}}$$ be a sequence of coordinatewise negatively associated random vectors taking values in a real separable Hilbert space with the k-th partial sum S, $${k \geqq 1}$$ . We provide conditions for the convergence of $${\sum_{n=1}^{\infty} \frac{1}{n} \, \mathbb{P}({\rm max}_{1 \leqq k \leqq n} \|S_k\| > \varepsilon n^{\alpha })}$$ and $${\sum_{n=1}^{\infty} \frac{{\rm log} n}{n} \, \mathbb{P}({\rm max}_{1 \leqq k \leqq n} \|S_k\| > \varepsilon n^{\alpha})}$$ for all $${\varepsilon > 0}$$ . The converses of these results are also discussed. [ABSTRACT FROM AUTHOR]

Density dependent Markov population processes with countably many types can often be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, provided that the total population size is large. They also exhibit diffusive stochastic fluctuations on a smaller scale about this deterministic path. Here, it is shown that the individuals in such processes experience an almost deterministic environment. Small groups of individuals behave almost independently of one another, evolving as Markov jump processes, whose transition rates are prescribed functions of time. In the context of metapopulation models, we show that 'individuals' can represent either patches or the individuals that migrate among the patches; in host-parasite systems, they can represent both hosts and parasites. [ABSTRACT FROM AUTHOR]

We study the convergence in probability in the non-standardM1Skorokhod topology of the Hilbert valued stochastic convolution integrals of the type to a process driven by a Lévy processL. In Banach spaces, we introduce strong, weak. and product modes of -convergence, prove a criterion for the -convergence in probability of stochastically continuous càdlàg processes in terms of the convergence in probability of the finite dimensional marginals and a good behavior of the corresponding oscillation functions, and establish criteria for the convergence in probability of Lévy driven stochastic convolutions. The theory is applied to the infinitely dimensional integrated Ornstein–Uhlenbeck processes with diagonalizable generators. [ABSTRACT FROM AUTHOR]

In this paper we derive properties of the least squares (or Karcher) mean of probability measures on the open cone Ω of positive definite matrices of some fixed dimension endowed with the trace metric that generalize known properties of the weighted least squares mean of finitely many positive definite matrices. Our approach is based on first defining the t -power mean of a probability measure as the unique fixed point of the contractive map X ∈ Ω ↦ ∫ Ω X # t Z μ ( d Z ) with respect to the Thompson metric, establishing its properties analogous to those of the power mean for a finite number of positive definite matrices, and showing the t -power means converge to the Karcher mean as t → 0 . We carry out this program first of all for the compactly supported probability measures and show that theory including the monotonicity extends to the general case. [ABSTRACT FROM AUTHOR]

Letbe a sequence of independent and identically distributedB-valued random variables, and set. Let,andq>0, and putWe strengthen the convergence rate for the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers in Banach space, by showing that, if and only ifand [ABSTRACT FROM PUBLISHER]

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\nu \in M^1([0,\infty [)$ be a fixed probability measure. For each dimension $p\in \mathbb{N}$, let $(X_n^{p})_{n\geq 1}$ be independent and identically distributed $\mathbb{R}^p$-valued random variables with radially symmetric distributions and radial distribution $\nu $. We investigate the distribution of the Euclidean length of $S_n^{p}:=X_1^{p}+\cdots + X_n^{p}$ for large parameters $n$ and $p$. Depending on the growth of the dimension $p=p_n$ we derive by the method of moments two complementary central limit theorems (CLTs) for the functional $\ S_n^{p}\ _2$ with normal limits, namely for $n/p_n \to \infty $ and $n/p_n \to 0$. Moreover, we present a CLT for the case $n/p_n \to c\in \, (0,\infty )$. Thereby we derive explicit formulas and asymptotic results for moments of radial distributed random variables on $\mathbb{R}^p$. All limit theorems are also considered for orthogonal invariant random walks on the space $\mathbb{M}_{p,q}(\mathbb{R})$ of $p\times q$ matrices instead of $\mathbb{R}^p$ for $p\to \infty $ and some fixed dimension $q$. [ABSTRACT FROM AUTHOR]

We develop the Baum-Katz theorem for sequences of coordinatewise negatively associated random vectors in Hilbert spaces. We also show that the concept of coordinatewise negative association is more general than the concept of negative association of Ko et al. []. Moreover, some related results still hold for this concept. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

For a double array of independent random elements { V, m ≥ 1, n ≥ 1} in a real separable Banach space, conditions are provided under which the weak and strong laws of large numbers for the double sums Σ Σ V, m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a self-similar Gaussian process denoted by with parameters and . This process generalizes the well-known fractional Brownian motion introduced by Mandelbrot and Van Ness [SIAM Rev. 10 (1968), no. 4, 422-437], the sub-fractional Brownian motion introduced by Bojdecki, Gorostiza and Talarczyket [Statist. Probab. Lett. 69 (2004), 405-419] and the Gaussian process introduced by Lei and Nualart [Statist. Probab. Lett. 79 (2009), 619-624] in order to obtain a decomposition in law of the bifractional Brownian motion. [ABSTRACT FROM AUTHOR]

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order p). In this paper, we give some new results of complete convergence in mean of order p and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces. [ABSTRACT FROM AUTHOR]

This paper derives some uniform convergence rates for kernel regression of some index functions that may depend on infinite dimensional parameter. The rates of convergence are computed for independent, strongly mixing and weakly dependent data respectively. These results extend the existing literature and are useful for the derivation of large sample properties of the estimators in some semiparametric and nonparametric models. [ABSTRACT FROM AUTHOR]

The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases. [ABSTRACT FROM AUTHOR]

This work develops almost sure convergence of negatively associated random vectors in Hilbert spaces. Extensions of a result in [4] are given. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]