Vector Valued Stochastic Processes III Projections and Dual Projections

1988 ◽  
pp. 93-122
Author(s):  
Nicolae Dinculeanu
Keyword(s):  
Stochastic Processes ◽  
Vector Valued

scholarly journals Toward a fundamental theorem of quantal measure theory

2012 ◽  
Vol 22 (5) ◽  
pp. 816-852 ◽  
Author(s):  
RAFAEL D. SORKIN
Keyword(s):  
Stochastic Processes ◽  
Path Integral ◽  
Fundamental Theorem ◽  
Measure Theory ◽  
Finite Lattice ◽  
Extension Problem ◽  
Integral Type ◽  
Causal Sets ◽  
Simple Event ◽  
Vector Valued

In this paper we address the extension problem for quantal measures of path-integral type, concentrating on two cases: sequential growth of causal sets and a particle moving on the finite lattice ℤn. In both cases, the dynamics can be coded into a vector-valued measure μ on Ω, the space of all histories. Initially, μ is just defined on special subsets of Ω called cylinder events, and we would like to extend it to a larger family of subsets (events) in analogy to the way this is done in the classical theory of stochastic processes. Since quantally μ is generally not of bounded variation, a new method is required. We propose a method that defines the measure of an event by means of a sequence of simpler events that in a suitable sense converges to the event whose measure we are seeking to define. To this end, we introduce canonical sequences approximating certain events, and we propose a measure-based criterion for the convergence of such sequences. Applying the method, we encounter a simple event whose measure is zero classically but non-zero quantally.

The Andersen-Jessen theorem revisited

1987 ◽  
Vol 102 (2) ◽  
pp. 351-361
Author(s):  
Klaus D. Schmidt
Keyword(s):  
Stochastic Processes ◽  
Almost Sure Convergence ◽  
Signed Measure ◽  
Convergence Theorems ◽  
Conditional Expectations ◽  
The Family ◽  
Set Function ◽  
Vector Valued

AbstractFor stochastic processes which are induced by a signed measure, the Andersen-Jessen theorem asserts almost sure convergence and yields the identification of the limit. This result has been extended to real and vector-valued stochastic processes which are induced by a finitely additive set function or a set function process. In the present paper, we study the structure of such induced stochastic processes in order to locate the Andersen-Jessen theorem and its extensions in the family of convergence theorems for martingales and their generalizations. As an application of these results, we also show that the Andersen-Jessen theorem and its extensions can be deduced from the convergence theorems for conditional expectations and positive supermartingales.

A GENERALIZATION OF FRACTAL INTERPOLATION STOCHASTIC PROCESSES TO HIGHER DIMENSIONS

Fractals ◽  
2001 ◽  
Vol 09 (04) ◽  
pp. 415-428 ◽  
Author(s):  
ROBERT MAŁYSZ
Keyword(s):  
Stochastic Processes ◽  
Higher Dimensions ◽  
Upper Bounds ◽  
Fractal Interpolation ◽  
Original Process ◽  
Stationary Increments ◽  
Fractal Interpolation Functions ◽  
Self Similar ◽  
Vector Valued ◽  
Interpolation Functions

We generalize the notion of fractal interpolation functions (FIFs) to stochastic processes. We prove that the Minkowski dimension of trajectories of such interpolations for self-similar processes with stationary increments converges to 2-α. We generalize the notion of vector-valued FIFs to stochastic processes. Trajectories of such interpolations based on an equally spaced sample of size n on the interval [0,1] converge to the trajectory of the original process. Moreover, for fractional Brownian motion and, more generally, for self-similar processes with stationary increments (α-sssi) processes, upper bounds of the Minkowski dimensions of the image and the graph converge to the Hausdorff dimension of the image and the graph of the original process, respectively.

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