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

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.