A seminar by Professor Katsuto Tanaka from Hitotsubashi University, Tokyo
Title: Brownian motion, the Fredholm determinant, and time series analysis
Abstract: The title of my talk comes from that of my book to be published from CUP in 2025. I consider distributions of quadratic functionals of Brownian motion (Bm), sometimes extended with linear or bilinear functionals, or ratios of those functionals. For this purpose, we attempt to derive the associated characteristic function (c.f.). For purely quadratic functionals of Bm, the theorem established by Anderson and Darling tells us that the derivation of the c.f. of such statistics results in computing the Fredholm determinant (FD). If linear or bilinear functionals are added to the quadratic Brownian functional, we need to deal with the resolvent. The notion of the FD and resolvent originates from the theory of integral equations of Fredholm type. Once the c.f. is obtained, the distribution function can be computed by inverting it numerically. I discuss how to compute the FD and the resolvent together with various statistical applications.
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