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A random walk of a particle in ${R}^d$ is considered. The weak convergence of various transformations of trajectories of random flights with Poisson switching times was studied by Davydov and Konak...

The classical Fatou lemma states that the lower limit of a sequence of integrals of functions is greater than or equal to the integral of the lower limit. It is known that Fatou's lemma for a seque...

We treat a fairly broad class of financial models that includes markets with proportional transaction costs. We consider an investor with cumulative prospect theory preferences and a nonnegativity ...

Sufficient conditions for the weak convergence of the distributions of the random variables $(1-x)\xi_x$ as $x\to1-$ to the limiting gamma-distribution are put forward. The random variable $\xi_x$ ...

We consider a one-parameter regression model with a regressor composed of order statistics that do not depend on regression errors. We calculate an empirical bridge of residuals and prove its weak convergence to a centered Gaussian process. We study statistical criteria based on the convergence.

The Bernoulli sieve is a random allocation scheme obtained by placing independent points with the uniform [0,1] law into the intervals made up by successive positions of a multiplicative random walk with factors taking values in the interval (0,1). Assuming that the number of points is equal to n we investigate the weak convergence, as n tends to infinity, of finite-dimensional distributions of the number of empty intervals within the occupancy range. A new argument enables us to relax the constraints imposed in previous papers on the distribution of the factor of the multiplicative random walk., Comment: submitted, 26 pages

Smirnov [Trudy Mat. Inst. Steklov., 25 (1949), pp. 3--60 (in Russian); Amer. Math. Transl., 67 (1952) (in English)] derived four limit types of distributions for linearly normalized central order statistics under the weak convergence. In this paper we investigate the possible limit distributions of the $k$th upper order statistics with central rank using monotone regular norming sequences and obtain 13 possible types.

Fatou's lemma states under appropriate conditions that the integral of the lower limit of a sequence of functions is not greater than the lower limit of the integrals. This paper describes similar inequalities when, instead of a single measure, the functions are integrated with respect to different measures that form a weakly convergent sequence.

We prove the asymptotic independence of the empirical process $\alpha_n = \sqrt{n}( F_n - F)$ and the rescaled empirical distribution function $\beta_n = n (F_n(\tau+\frac{\cdot}{n})-F_n(\tau))$, where $F$ is an arbitrary cdf, differentiable at some point $\tau$, and $F_n$ the corresponding empricial cdf. This seems rather counterintuitive, since, for every $n \in N$, there is a deterministic correspondence between $\alpha_n$ and $\beta_n$. Precisely, we show that the pair $(\alpha_n,\beta_n)$ converges in law to a limit having independent components, namely a time-transformed Brownian bridge and a two-sided Poisson process. Since these processes have jumps, in particular if $F$ itself has jumps, the Skorokhod product space $D(R) \times D(R)$ is the adequate choice for modeling this convergence in. We develop a short convergence theory for $D(R) \times D(R)$ by establishing the classical principle, devised by Yu. V. Prokhorov, that finite-dimensional convergence and tightness imply weak convergence. Several tightness criteria are given. Finally, the convergence of the pair $(\alpha_n,\beta_n)$ implies convergence of each of its components, thus, in passing, we provide a thorough proof of these known convergence results in a very general setting. In fact, the condition on $F$ to be differentiable in at least one point is only required for $\beta_n$ to converge and can be further weakened.

The main result of this paper is a theorem of portmanteau type for pseudoweak convergent sequences of capacity functionals for a sequence of random closed sets. For that purpose the classical Lebesgue integral had been substituted with a more general one, known as general pseudo-integral, and there is introduced the pseudoweak convergence of capacity functionals. A connection between weak convergence of a sequence of probability measures induced by the sequence of random closed sets and convergence of pseudo-integral with respect to the corresponding sequence of capacity functionals is given.

A modified discrete limit theorem in the sense of the weak convergence of probability measures in the space of analytic functions for the Matsumoto zeta-function is proved.

For the independent polynomial allocation scheme of particles of two types we consider the numbers of cells containing given numbers of particles of each type. A theorem on the weak convergence of the vector formed by these numbers to the random vector with independent Poisson components is proved.

We extend Marcinkievicz–Zygmund strong laws of large numbers for martingales to weakly dependent random variables with values in smooth Banach spaces. The conditions are expressed in terms of conditional expectations. In the case of Hilbert spaces, we show that our conditions are weaker than optimal ones for strongly mixing sequences (which were previously known for real-valued variables only). As a consequence, we give rates of convergence for Cramer–von Mises statistics and for the empirical estimator of the covariance operator of a Hilbert-valued autoregressive process.

The nonlinear SDE of McKean–Vlasov type in the absence of external fields is considered. First, the existence and the uniqueness of the equation solution are proved. Next, the existence and the uniqueness of the stationary solution at the class of probability with fixed expectation are proved. Also, weak convergence to invariant probability is proved.

Characterizations of the class of Banach spaces isomorphing to the space $c_0$, as well as to the class of Banach spaces not containing $l_\infty^n$'s uniformly, are obtained in terms of almost surely unconditional convergence of weakly sub-Gaussian random series. Under almost surely unconditional convergence of random series, convergence of all permutations on the same set of full probability is understood. The questions of almost surely unconditional and weak absolute convergence in the spaces isomorphing to $c_0$ are investigated as well.

We prove a large deviations upper bound for the kernel estimator of the mode when this one corresponds to a strict maximum of a probability density function. Surprisingly, the convergence rate we obtain is faster than one could expect in view of the weak convergence rate of the studied estimator.

The main objective of this paper consists in discussing the concept of weak solutions of a certain type of backward stochastic differential equations. Using weak convergence in the Meyer--Zheng topology, we shall give a general existence result. The terminal condition H depends in functional form on a driving cadlag process X, and the coefficient f depends on time t and in functional form on X and the solution process Y. The functional $f(t,x,y),(t,x,y)\in [0,T]\times D([0,T];{\bf R}^{d+m})$ is assumed to be bounded and continuous in $(x,y)$ on the Skorokhod space $D([0,T]\,;{\bf R}^{d+m})$ in the Meyer--Zheng topology. By several examples of Tsirelson type, we will show that there are, indeed, weak solutions which are not strong, i.e., are not solutions in the usual sense. We will also discuss pathwise uniqueness and uniqueness in law of the solution and conclude, similar to the Yamada--Watanabe theorem, that pathwise uniqueness and weak existence ensure the existence of a (uniquely determined) strong so...

We consider the problem of a posteriori change-point detection for a sequence of independent identically distributed random variables. We propose to use d-risks instead of error of the first type and error of the second type. We construct an asymptotically optimal test minimizing one d-risk and guaranteeing another.

One-dimensional stochastic equations are considered whose coefficients depend on a small parameter. Necessary and sufficient conditions are obtained for the weak convergence of their solutions to the solution of the stochastic equation containing local time of an unknown process.

This paper provides necessary and sufficient conditions for weak convergence of the distributions of sums of independent random variables to normal and Poisson distributions. The conditions are given in terms of the first through fourth moments of the truncated components and some condition guaranteeing the asymptotic equality between the distribution of the sums mentioned above and the distribution of the sums of the truncated components. The theorems proved in this work are similar to the well-known criteria for normal and Poisson convergencies.

In this paper, we consider the asymptotic behavior of convex rearrangements for regularizations of paths of Gaussian processes with stationary increments, and we use the concentration principle to prove the almost sure convergence of these rearrangements to a nonrandom convex line, the so-called Lorentz curve, corresponding to the standard Gaussian law. Moreover, we obtain the same type of result for the Gaussian bridges of such processes. We also discuss relations with the recent results of Azais and Wschebor about the almost sure weak convergence of oscillations of Gaussian processes. As an application of our basic theorem we prove a theorem of Baxter type for p-variations of the paths and introduce a new family of consistent estimators of the fractal index.

In this paper we consider stationary sequences under the validity of an extension of Leadbetter's condition $D(u_n)$. For these sequences we prove that, if $\{k_n\}$ is a nondecreasing integer sequence satisfying $\lim_{n\rightarrow +\infty}k_{n+1}/ k_{n}=r\ge 1$, then the limit law for the maximum of the first $k_n$ variables is a max-semistable law. This generalizes the corresponding result for sequences of independent identically distributed random variables of Grinevich [{\em Theory Probab. Appl.}, 38 (1993), pp.~640--650] and the extremal types theorem of Leadbetter [{\em Z. Wahrsch. Verw. Gebiete}, 28 (1974), pp.~289--303]. We also prove that the limiting behavior of this maximum can be inferred from the limiting behavior of the corresponding maximum of the associated independent sequence, and we extend the well-known notion of extremal index. An illustrative example is given.

We consider the functional $T_n=(S_1^2+\cdots+S_n^2)/(nV_n^2)$ derived from a sequence $\{X_n\}_{n\ge 1}$ of independent identically distributed random variables, where $S_k=X_1+\cdots+X_k$, $V_n^2=X_1^2+\cdots+X_n^2$. Let~G be the distribution function of the random variable $\int_{0}^{1}W^2(t)\,dt$, where~$W(t)$, $t\in [0,1]$, is a Wiener process. We show that the distribution function $T_n$ weakly converges to~G as $n\to\infty$ if and only if the distribution function of the random variable~$X_1$ belongs to the attraction domain of the normal law and~${\mathbf{E} X_1=0}$.

A sufficient condition is given for the a.s. unconditional convergence of Gaussian series in Banach spaces with unconditional bases not containing $l^n_\infty$'s uniformly. By the a.s. unconditional convergence of random series we understand the convergence of all rearrangements of the series on the same set of total probability.

This paper gives necessary and sufficient conditions for the weak convergence of joint distributions of sample quantiles constructed from samples of random sizes. The limit distributions are described.

Estimates are found for the magnitude of overshoot, by a sequence of random variables, over an arbitrary boundary. If the sequence increments satisfy a so-called condition of asymptotic homogeneity and the boundary is asymptotically ``smooth," then the occurrence of the weak convergence to a limit shape (as the boundary is sent away) is established for the distribution of the overshoot value. As an application, we obtain a uniform (over the class of distributions) basic renewal theorem and determine the asymptotics of the average time of crossing a curvilinear border by the trajectories of asymptotically homogeneous Markov chains.

One-dimensional stochastic equations are considered whose coefficients depend on a small parameter in an irregular way. In terms of coefficients, necessary and sufficient conditions are obtained for the weak convergence of their solutions to the solution of the stochastic equation. Examples are given.

Inverse estimation concerns the recovery of an unknown input signal from blurred observations on a known transformation of that signal. The estimators considered in this paper are based on a regularized inverse of the transformation involved, employing a Hilbert space set-up. We focus on properties related to weak convergence. It is shown that linear functionals can be efficiently estimated in the H\'{a}jek--LeCam sense, provided they remain restricted to a suitable class. Outside this class, rates different from $\sqrt{n}$ are possible. By way of an example we present the ``convolution theorem' for a deconvolution.

In this paper a stochastic control model is studied that admits a diffusion approximation. In the prelimit model the disturbances are given by noise processes of various types: additive stationary noise, rapidly oscillating processes, and discontinuous processes with large intensity for jumps of small size. We show that a feedback control that satisfies a Lipschitz condition and is $\delta$-optimal for the limit model remains $\delta$-optimal also in the prelimit model. The method of proof uses the technique of weak convergence of stochastic processes. The result that is obtained extends a previous work by the authors, where the limit model is deterministic.

The paper proposes nonparametric estimates for exponential cumulative intensity functions based on the generalized model of random censorship from the right. For respectively normed estimates, the results on strong approximationof them by a sequence of square integrable orthogonal martingales and on the weak convergence to the limit Gaussian process are proved.

The shift-compactness of random sums $S_{N_n}^{(n)}$, $S_k^{(n)}=X_{n,1}+\cdots +X_{n,k}$, of independent random variables is investigated under the assumptions that in each sum the summands and their number $N_n$ are independent and that the summands satisfy the condition of uniform asymptotic negligibility in the form $$ \max_{1\le k\le N_n}\bP\big\{|X_{n,k}|\ge \ve\big\}\longrightarrow 0 $$ in probability for each $\ve >0$. Some necessary and sufficient conditions are given for the weak compactness of random sums $S_{N_n}^{(n)}-A_n$, and the form of centering constants $A_n$ is described.