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

Chii-Ruey Hwang
1982 Transactions of the American Mathematical Society  
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/(x, y) on which some integrability conditions are imposed. Let T denote the integral operator with kernel/. When n i.i.d. generators (Xk,Yk) are connected together to form the configuration space 6" via the regularity (EQUAL, LINEAR), i.e. "conditioning" onïH, = Yk for 1 «s k < n, an approximate identity is used to define the
more » ... ty controlled probability on Qn. The probabilistic effect induced by the regularity conditions on some fixed subconfiguration of a larger configuration ß" is described by its corresponding marginal probability within S". 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*. When / is bivariate normal, the eigenvalue problem is solved explicitly. The process determined by the limiting marginal probabilities is strictly stationary and Markovian.
doi:10.1090/s0002-9947-1982-0670919-9 fatcat:jb45rlqddzglhkwhxkr2vcwjhu