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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] Y.-S. Chow and L.-D. Wu, On some limit theorems of regularity controlled probabilities, R. P. A. No. 108, Brown Univ., Providence, R.I., 1981.
  • [2] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
  • [3] Avner Friedman, Foundations of modern analysis, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0275100
  • [4] U. Grenander, Pattern synthesis, Springer-Verlag, New York-Heidelberg, 1976. Lectures in pattern theory, Vol. 1; Applied Mathematical Sciences, Vol. 18. MR 0438838
  • [5] -, Solve the 2nd limit problem in metric pattern theory, R. P. A. No. 83, Brown Univ., Providence, R.I., 1979.
  • [6] -, Thirty-one problems in pattern theory, R. P. A. 85, Brown Univ., Providence, R.I., 1980.
  • [7] Ulf Grenander, Regular structures, Applied Mathematical Sciences, vol. 33, Springer-Verlag, New York-Berlin, 1981. Lectures in pattern theory. Vol. III. MR 619018
  • [8] C.-R. Hwang and L. Andersson, On conditioning along the diagonal, R. P. A. No. 78, Brown Univ., Providence, R.I., 1979.
  • [9] C.-R. Hwang, On the 2nd limit problem in metric pattern theory, working paper No. 2, Institute of Math., Academia Sinica, 1979.
  • [10] Lambert H. Koopmans, Asymptotic rate of discrimination for Markov processes, Ann. Math. Statist. 31 (1960), 982–994. MR 0119368
  • [11] C. Plumeri, Conditioning by equality when connection type is linear, R. P. A. No. 86, Brown Univ., Providence, R.I., 1980.
  • [12] -, Probability measures on regular structures, R. P. A. No. 100, Brown Univ., Providence, R.I., 1981.
  • [13] P. P. Zabreyko et al., Integral equations, Noordhoff, Gröningen, 1975.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60B99, 68G10

Retrieve articles in all journals with MSC: 60B99, 68G10


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