Probability Theory on Infinite Dimensional Spaces

The remarkable link between the structure of the white noise and that of the infinite dimensional rotation group has been exemplified by various approaches in probability theory and harmonic analysis. Such a link naturally becomes more intricate as the dimension of the time-parameter space of the white noise increases. One of the powerful method to illustrate this situation is to observe the structure of certain subgroups of the infinite dimensional rotation group that come from the diffeomorphisms of the time-parameter space, that is the time change. Indeed, those subgroups would shed light on the probabilistic meanings hidden behind the usual formal observations. Moreover, the subgroups often describe the way of dependency for Gaussian random fields formed from the white noise as the time-parameter runs over the basic parameter space.The main purpose of this note is to introduce finite dimensional subgroups of the infinite dimensional rotation group that have important probabilistic meanings and to discuss their roles in probability theory. In particular, we shall see that the conformal invariance of white noise can be described in terms of the conformal group which is a finite dimensional Lie subgroup of the infinite dimensional rotation group.

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In memoriam: Czesław Ryll-Nardzewski’s contributions to probability theory

Probability and Mathematical Statistics ◽

10.19195/0208-4147.37.1.1 ◽

2018 ◽

Vol 37 (1) ◽

pp. 1-20

Author(s):

Tomasz Rolski ◽

Wojbor A. Woyczyński

Keyword(s):

Probability Theory ◽

Power Series ◽

Ergodic Theory ◽

Point Processes ◽

Conditional Distributions ◽

Random Series ◽

Infinite Dimensional ◽

Random Power Series ◽

In Memoriam

IN MEMORIAM: CZESŁAW RYLL-NARDZEWSKI’S CONTRIBUTIONS TO PROBABILITY THEORYIn this paper we review contributions of late Czesław Ryll-Nardzewski to probability theory. In particular, we discuss his papers on point processes, random power series, random series in infinite-dimensional spaces, ergodic theory, de Finetti’s exchangeable sequences, conditional distributions and applications of the Kuratowski–Ryll-Nardzewski theorem on selectors.

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