Conditioning by $\langle$equal, linear$\rangle$
HTML articles powered by AMS MathViewer
- by Chii-Ruey Hwang PDF
- Trans. Amer. Math. Soc. 274 (1982), 69-83 Request permission
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
-
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.
- Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
- Avner Friedman, Foundations of modern analysis, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1970. MR 0275100
- U. Grenander, Pattern synthesis, Applied Mathematical Sciences, Vol. 18, Springer-Verlag, New York-Heidelberg, 1976. Lectures in pattern theory, Vol. 1. MR 0438838 —, Solve the 2nd limit problem in metric pattern theory, R. P. A. No. 83, Brown Univ., Providence, R.I., 1979. —, Thirty-one problems in pattern theory, R. P. A. 85, Brown Univ., Providence, R.I., 1980.
- Ulf Grenander, Regular structures, Applied Mathematical Sciences, vol. 33, Springer-Verlag, New York-Berlin, 1981. Lectures in pattern theory. Vol. III. MR 619018 C.-R. Hwang and L. Andersson, On conditioning along the diagonal, R. P. A. No. 78, Brown Univ., Providence, R.I., 1979. C.-R. Hwang, On the 2nd limit problem in metric pattern theory, working paper No. 2, Institute of Math., Academia Sinica, 1979.
- Lambert H. Koopmans, Asymptotic rate of discrimination for Markov processes, Ann. Math. Statist. 31 (1960), 982–994. MR 119368, DOI 10.1214/aoms/1177705671 C. Plumeri, Conditioning by equality when connection type is linear, R. P. A. No. 86, Brown Univ., Providence, R.I., 1980. —, Probability measures on regular structures, R. P. A. No. 100, Brown Univ., Providence, R.I., 1981. P. P. Zabreyko et al., Integral equations, Noordhoff, Gröningen, 1975.
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 274 (1982), 69-83
- MSC: Primary 60B99; Secondary 68G10
- DOI: https://doi.org/10.1090/S0002-9947-1982-0670919-9
- MathSciNet review: 670919