A Factorization problem and the problem of predicting non-stationary vector-valued stochastic processes

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.

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Fixed-Rate Zero-Delay Source Coding for Stationary Vector-Valued Gauss-Markov Sources

2018 Data Compression Conference ◽

10.1109/dcc.2018.00034 ◽

2018 ◽

Cited By ~ 3

Author(s):

Photios A. Stavrou ◽

Jan Ostergaard

Keyword(s):

Source Coding ◽

Fixed Rate ◽

Vector Valued ◽

Stationary Vector ◽

Markov Sources

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The Andersen-Jessen theorem revisited

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100067360 ◽

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.

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