Asymptotically invariant sampling and averaging (Q1613611)

The following important theorem is proved for an \(S\)-valued (\(S\) is a Polish space) shift-stationary process \(X\) on \(T\), where \(T\) is a semigroup of \(\mathbb R^d\) for some \(d \in \mathbb N\): If \((\tau _1, \tau _2, \dots{})\) are independent sequences in \(T\) with asymptotically invariant distributions and \(\xi _k^n=X(\tau _k^n)\), \(\xi ^n =(\xi _1^n,\xi _2^n, \dots{})\), then \(\xi ^n\) is asymptotically exchangeable: \(\xi ^n @>d>> \xi \) in \(S^\infty \) for some exchangeable sequence in \(S\). Further, let \(\mu _i,\;i=1,2,\dots{}\), be asymptotically invariant distributions on \(T^\infty \) and \(\mu _i\delta ^\infty _x\) the associated random measures on \(S^\infty \) defined through averaging of \(X\): \(\mu _i\delta ^\infty _x B=\int 1_B(X_{t_1},X_{t_1},\dots{}\mu _i(dt_1, dt_1,\dots))\), \(B\in \mathcal S^\infty \); \(\mathcal S\) is the Borel \(\sigma \)-field and \(\mathcal S^\infty \) the corresponding infinite product \(\sigma \)-field. The proof of the above theorem is based on the mean ergodic theorem for \(\mu _i\delta ^\infty _x\) stated in terms of weak convergence in probability \(@>\text{wP}>>\) in the space \(\mathcal M_1(S)\): For \(X\) as above, there exists some random distribution \(\rho \) on \(S\) such that \(\mu_i\delta ^\infty _x @>\text{wP}>>\rho ^\infty \). If \((\mu_1, \mu _2,\dots{})\) form an absolutely continuous scaling family, the ergodic theorem can be strengthened to a pointwise a.s. version applicable to Poisson and Bernoulli sampling. The consequences of the main theorem are discussed, in particular its relation to the famous de Finetti and Ryll-Nardzewski theorem on finite exchangeable sequences as well as to other theorems included in the author's book ``Foundations of modern probability'' (1997; Zbl 0892.60001). Finally, an extension of the main theorem for a stationary \(X\) is proposed and proved and other related results are given.