On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in LEP

The weak convergence of certain functionals of a sequence of stochastic processes is investigated. The functionals under consideration are of the form f_φ(x) = ∫ φ (t, x(t))μ(dt). The main result is as follows: If a sequence $\text{{ξ}{}_{\mathrm{n}}\text{:}\text{n}\text{∈}\text{Z}$ is weakly tight in a certain sense, and, in addition, the finite dimensional distributions of the processes converge weakly, then this implies weak convergence of the functionals (f_φ1(ξ_n),…, f_φm(ξ_n)) to (f_φ1(ξ₀),…, f_φm(ξ₀)). Necessary and sufficient conditions for weak tightness are stated and applications of the results to the case of L^E_p-valued stochastic processes are given, ln particular it is shown that the usual tightness condition for weak convergence of such processes can be considerably weakened.     

On the weak convergence of stochastic processes without discontinuities of the second kind

Summary Subspaces D , > 0, of D[0, 1] are defined and given complete metrics d which are stronger than the Prokhorov metric. The spaces (D d ) are shown to be separable, and their pre-compact subsets are characterized. A condition which is known to guarantee weak pre-compactness of sets of probability measures over D[0, 1] is shown to also guarantee weak pre-compactness of probability measures over D for appropriate values of . Applications are made to the weak convergence of measures induced by stochastic processes, and some examples are included.     

Variational derivatives and p-gradients of functionals on spaces of continuously differentiable functions

The purpose of this paper is to introduce and study a new type of derivative – the variational gradient – for a functional on Cⁿ[a, b]. Local and global versions of this concept are analyzed. This notion provides a natural approach to variational derivatives on Cⁿ[a, b] under rather mild smoothness assumptions on the functional. When applied in the context of the Calculus of Variations, the notion of the variational gradient captures the natural boundary conditions (as well as the Euler-Lagrange equations) under weaker smoothness assumptions than those usually required using Gǎteaux variations. Conditions are established for the existence of the variational derivative and an integral representation for the Gǎteaux variation in terms of the variational derivative is derived. Conditions for the variational derivative to be differentiable are also established.     

On weak convergence of stocastic processes with Lusin path spaces

Under the assumption that a sequence of stochastic processes has paths in a Lusin function space we can prove the following. If convergence in the path space implies stochastic convergence, then tightness and convergence of the finite dimensional distributions of the stochastic processes are sufficient for weak convergence. The result in many cases implies a unification of the weak convergence proof. Demonstrably, such cases are C, D, Lip, L_p and , the space of distribution functions of finite measures.     

On weak convergence of solutions of one-dimensional stochastic differential equations

The main purpose of the present paper is to study weak convergence of solutions of stochastic differential equations     

On the invariance principle for differentiable statistical functionals

Sup-norm differentiability of a statistical functional T is shown to be a sufficient condition for invariance principles for T. Application of the result to repeated significance testing is sketched.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.     

General convergence results for stochastic approximations via weak convergence theory

The paper treats general convergence conditions for a class of algorithms for finding the minima of a function f(x) when f(x) is of unknown (or partly unknown) form, and when only noise corrupted observations can be taken. Such problems occur frequently in adaptive processes, and in many applications to statistics and estimation. The algorithms are of the stochastic approximation type. Several forms are dealt with—for estimation in either discrete or continuous time, with and without side constraints, and with or without periodic search renewal. The algorithms can be considered as sequential Monte Carlo methods for systems optimization. The innovations partly concern the method of proof. However, an interesting “constrained” and “renewed” algorithm is also considered. By using ideas from the theory of weak convergence of probability measures, we can get relatively short proofs, under much weaker conditions than heretofore required. For example, the noise can be correlated, and there are fewer restrictions on the step size. Furthermore, the nature of the method permits generalizations to more abstract cases (which occur for example, if we are optimizing a distributed parameter system). The results can be extended in many directions and variations of the technique can be used to get bounds on rates of convergence. Special forms of the method can be applied to many well-known “adaptive” procedures.     

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.     

On conditional weak convergence

In this paper we discuss a number of technical issues associated with conditional weak convergence. The main modes of convergence of conditional probability distributions areuniform, probability, andalmost sure convergence in the conditioning variable. General results regarding conditional convergence are obtained, including details of sufficient conditions for each mode of convergence, and characterization theorems for uniform conditional convergence.