Elements of a function analytic approach to probability. (English) Zbl 1176.65011
Summary: Three hundred-plus years of successful theoretical development and application of probability theory provide sufficient justification for it as the mathematical context in which to analyze the uncertainty in the performance of engineering and scientific systems. In this document, we propose a joint probabilistic and deterministic function analytic approach as the means for the development of advanced techniques that feature a strong connection between classical deterministic and probabilistic methods. We know of no other means to achieve simultaneous, balanced approximations across these two constituents. We present foundational materials on the general approach to particular aspects of functional analysis, which are relevant to probability, and emphasize the common elements it shares, and the close connections it provides, to various classical deterministic mathematical analysis elements. Finally, we describe how to use the joint approach as a means to augment deterministic analysis methods in a particular Hilbert space context, and thus enable a rigorous framework for commingling deterministic and probabilistic analysis tools in an application setting.
MSC:
| 65C60 | Computational problems in statistics (MSC2010) |
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