Conditioning by equal, linear

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 on which some integrability conditions are imposed. Let denote the integral operator with kernel . When i.i.d. generators are connected together to form the configuration space via the regularity , i.e. "conditioning" on for , an approximate identity is used to define the regularity controlled probability on . The probabilistic effect induced by the regularity conditions on some fixed subconfiguration of a larger configuration is described by its corresponding marginal probability within . When 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 and . When 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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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0670919-9

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