Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Conditioning by $ \langle $equal, linear$ \rangle $

Author: Chii-Ruey Hwang
Journal: Trans. Amer. Math. Soc. 274 (1982), 69-83
MSC: Primary 60B99; Secondary 68G10
MathSciNet review: 670919
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Abstract: We deal with a limit problem of regularity controlled probabilities in metric pattern theory. The probability on the generator space is given by a density function $ f(x,y)$ on which some integrability conditions are imposed. Let $ T$ denote the integral operator with kernel $ f$. When $ n$ i.i.d. generators $ ({X_k},{Y_k})$ are connected together to form the configuration space $ {\mathcal{C}_n}$ via the regularity $ \left\langle {{\text{EQUAL}},{\text{LINEAR}}} \right\rangle $, i.e. "conditioning" on $ {X_{k + 1}} = {Y_k}$ for $ 1 \leqslant k < n$, an approximate identity is used to define the regularity controlled probability on $ {\mathcal{C}_n}$. The probabilistic effect induced by the regularity conditions on some fixed subconfiguration of a larger configuration $ {\mathcal{C}_n}$ is described by its corresponding marginal probability within $ {\mathcal{C}_n}$. When $ n$ goes to infinity in a suitable way, the above mentioned marginal probability converges weakly to a limit whose density can be expressed in terms of the largest eigenvalues and the corresponding eigenspaces of $ T$ and $ {T^ \ast }$. When $ f$ is bivariate normal, the eigenvalue problem is solved explicitly. The process determined by the limiting marginal probabilities is strictly stationary and Markovian.

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Keywords: Approximate identity, bivariate Gaussian, compact positive operator, conditioning, configuration, eigenvalue, eigenfunction, generator, Hermite polynomial, integral equation, Markovian, normal operator, pattern theory, regularity controlled probability, strictly stationary, weak convergence
Article copyright: © Copyright 1982 American Mathematical Society