A note on weak convergence of mean residual life of stationary mixing random variables

For a sequence of strictly stationary uniform or strong mixing we estimate the mean residual time of the marginal distribution from the first n observations. Under appropriate conditions it is shown that the estimate converges weakly to a well-defined Gaussian process even when the sample size is random.     

On weak convergence of random fields

We suggest simple and easily verifiable, yet general, conditions under which multi-parameter stochastic processes converge weakly to a continuous stochastic process. Connections to, and extensions of, R. Dudley’s results play an important role in our considerations, and we therefore discuss them in detail. As an illustration of general results, we consider multi-parameter stochastic processes that can be decomposed into differences of two coordinate-wise non-decreasing processes, in which case the aforementioned conditions become even simpler. To illustrate how the herein developed general approach can be used in specific situations, we present a detailed analysis of a two-parameter sequential empirical process.     

On weak convergence of random fields

We provide a characterization of compactness in the spaceD of functions of two variables defined on a unit square. The functions fromD have the property that their discontinuity points lie on smooth curves. Conditions for the tightness of probability measures inD and conditions for weak convergence of random fields with trajectories inD are derived. Vilnius Gediminas Technical University, Saulétekio 11; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp 169–184, April–June, 1999. Translated by R. Banys     

A note on processes with random stationary increments

When the correlation theory is considered for the processes with random stationary increments, Yaglom (1955) has developed the spectral representation theory. In this note, we complete this development by obtaining the inversion formula of the spectrum in terms of the structure function.     

The weak convergence of step processes to a homogeneous Poisson process with independent increments

One obtains conditions for the convergence of the step processes _n(t) to a homogeneous Poisson process with independent increments, where , t_nm_n is a sequence of partitions of the segment [0, 1], while , ..., nm_n is a sequence of series of independent random variables, satisfying the condition of limiting uniform smallness.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 293–314, 1986.     

随机过程序列部分和的收敛性的注记

讨论了随机过程序列部分和的弱收敛性，弱化了肖庆宪等人（1999）给出的条件并将结果推广到了混合序列的情形。     

关于相依随机变量加权和强收敛性的注记

放宽了随机变量的独立性的要求,在假设随机变量为Ψ-混合的条件下,讨论了加权和强收敛性。     

Functional weak convergence of partial maxima processes

For a strictly stationary sequence of nonnegative regularly varying random variables (X _n ) we study functional weak convergence of partial maxima processes \(M_{n}(t) = \bigvee _{i=1}^{\lfloor nt \rfloor }X_{i},\,t \in [0,1]\) in the space D[0, 1] with the Skorohod J ₁ topology. Under the strong mixing condition, we give sufficient conditions for such convergence when clustering of large values do not occur. We apply this result to stochastic volatility processes. Further we give conditions under which the regular variation property is a necessary condition for J ₁ and M ₁ functional convergences in the case of weak dependence. We also prove that strong mixing implies the so-called Condition \(\mathcal {A}(a_{n})\) with the time component.     

A remark on the weak convergence of processes in the Skorohod topology

A Skorohod representation type theorem is proved for the weak convergence of stochastic processes in the Skorohod topology. This allows the time changes arising from the Skorohod topology to be considered as stochastic processes. While thenth time change processA _t ⁿ is not adapted to thenth filtration ( _t ⁿ ) _t0, it is possible to choose the processesA ⁿ such that they are adapted to, where, where _n is a sequence of constants decreasing to 0 asn tends to .Supported in part by NSF Grant No. DMS-9103454.     

Bivariate step processes and random evolutions

A random evolution process constructed from regular step processes with a common state space and indexed on an evolution rule space is shown to be a regular step process on the product space. Conversely, it is shown that under mild conditions, any regular step process on a product space is equivalent to a random evolution process. Conditions are given on the cardinality of the spaces and on the parameters of the process that are sufficient for the process to have various recurrence and ergodicity properties. Applications to birth-death processes are given.     

A note on the convergence of sequences of conditional expectations of random variables

Summary We disprove two theorems on the convergence of sequences of conditional expectations of random variables in [1] by providing a counterexample.     

A note on uniform exponential convergence of sample average approximation of random functions

Shapiro and Xu (2008) [17] investigated uniform large deviation of a class of Hölder continuous random functions. It is shown under some standard moment conditions that with probability approaching one at exponential rate with the increase of sample size, the sample average approximation of the random function converges to its expected value uniformly over a compact set. This note extends the result to a class of discontinuous functions whose expected values are continuous and the Hölder continuity may be violated for some negligible random realizations. The extension entails the application of the exponential convergence result to a substantially larger class of practically interesting functions in stochastic optimization.