Looking at a stochastic process as a function plucked at random may be interesting in its own right, but there is a statistical reason for doing so: a large class of limit theorems for functions of partial sums of a sequence of random variables or for statistics which are functions of empirical processes may be derived. To achieve such practicality, we are led to ask a number of requirements of a theory of probability measures on a function space. We enumerate a few. (i) To begin with, our function space S must contain the realizations of processes of interest to us and should contain those which are defined in their simplest forms with empirical observations. (ii) The a-field aJ of subsets of S over which our probability measures are defined must contain events whose probability we wish to know. Some of these events may be defined by topological or analytical qualifications; for example, the classes of continuous functions in S or integer valued functions in S. (iii) Processes under investigation must be measurable functions from the underlying probability space over which they are defined; that is, they must induce probability measures on -. Moreover, we should have convenient rules in order to determine whether a process is measurable. (iv) Each real valued and measurable function h on S transforms a stochastic process or random function X into a statistic or random variable h(X). Statistics of interest to us should be so induced by measurable functions of stochastic processes. In addition, we must have criteria, preferably easy to apply, to verify the measurability of real functions h on S and to determine if h(X) is a random variable. (v) For a sequence of stochastic processes {X,,} a mode of convergence should be defined so that we may infer, for a large class of functions {h}, that the statistic h(X5) converges weakly to h(Xo); that is, the distribution function of h(X") converges to that of h(Xo) at each continuity point of the latter. In addition, we need facile standards to determine for a function h when such convergence holds.

background

/pdf/weak-convergence-of-stochastic-processes-with-several-51rq42vn2a.pdf

A Criterion for Weak Convergence of Measures with an Application to Convergence of Measures on $D[0, 1]$.

Weak convergence of stochastic processes with several parameters

A new topology (called $S$) is defined on the space $D$ of functions $x: [0,1] \to R^1$ which are right-continuous and admit limits from the left at each $t > 0$. Although $S$ cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorohod's topologies $J_1$ and $M_1$. In particular, on the space $P(D)$ of laws of stochastic processes with trajectories in $D$ the topology $S$ induces a sequential topology for which both the direct and the converse Prohorov's theorems are valid, the a.s. Skorohod representation for subsequences exists and finite dimensional convergence outside a countable set holds.

methods

/pdf/a-non-skorohod-topology-on-the-skorohod-space-363rblp08j.pdf

A Non-Skorohod Topology on the Skorohod Space

We apply the method of Lyapunov functions to differential operators associated with diffusion processes in Hilbert spaces in order to ensure existence and uniqueness of stationary distributions and their attractivity. We restrict matters to the case of coefficients that are everywhere defined and derive our results from more general ones on completely regular Radon spaces. We also give an example of nonuniqueness and apply these results to systems of scalar, interacting, stochastic differential equations

Lyapunov-type conditions for stationary distributions of diffusion processes on hilbert spaces

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients, necessary and sufficient conditions for the finite dimensional convergence to an $\alpha $-stable Levy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $\alpha \le 1$. In the functional setting, we complement the existing results on $M_1$-convergence, obtained for linear processes with nonnegative coefficients by Avram and Taqqu (Ann Probab 20:483–503, 1992) and improved by Louhichi and Rio (Electr J Probab 16(89), 2011), by proving that in the general setting partial sums of linear processes are convergent on the Skorokhod space equipped with the $S$ topology, introduced by Jakubowski (Electr J Probab 2(4), 1997).

/pdf/functional-convergence-of-linear-processes-with-heavy-tailed-j3jtc9a7yi.pdf

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

The first part of the paper contains a survey of the present state of research in the general problems of convergence of stochastic processes. An important place is given to theorems on weak convergence of distributions on metric spaces. Then a different approach is proposed to the study of convergence of distributions of functionals on stochastic processes, which is connected with the approximation of the trajectories of a process by some family of functions. We think of approximation in terms of the nearness of the functionals in question. Using this approach we obtain all the main results on convergence in specific function spaces that are known at present. These results are obtained in their most general form without the requirement that the limiting processes should belong to the space under discussion. New limit theorems are also obtained and among them theorems for processes with discontinuities of the second kind and others.

http://iopscience.iop.org/article/10.1070/RM1972v027n01ABEH001362/pdf

The convergence of distributions of functionals on stochastic processes

/pdf/a-criterion-for-weak-convergence-of-measures-with-an-c6412lr3f3.pdf

A Criterion for Weak Convergence of Measures with an Application to Convergence of Measures on $D[0, 1]$.

Citations

Weak convergence of stochastic processes with several parameters

A Non-Skorohod Topology on the Skorohod Space

Lyapunov-type conditions for stationary distributions of diffusion processes on hilbert spaces

Functional Convergence of Linear Processes with Heavy-Tailed Innovations

The convergence of distributions of functionals on stochastic processes

Related Papers (5)

On a Criterion of Weak Convergence

On the Weak Convergence of Probability Measures

Weak Convergence of Probability Measures.

Convergence of Probability Measures

Weak Convergence of Probability Measures Revisited