Ornstein-Chacon ergodic theorem

2020 Mathematics Subject Classification: Primary: 37A30 Secondary: 47A35 [MSN][ZBL]

Let $(W,\mu)$ be a space with a $\sigma$-finite measure and let $T$ be a positive linear operator on $L_1(W,\mu)$ with $L_1$-norm $\Vert T\Vert\leq1$. If $f,g\in L_1(W,\mu)$ and $g\geq0$ almost everywhere, then the limit

$$\lim_{n\to\infty}\frac{\sum_{k=0}^nT^kf(w)}{\sum_{k=0}^nT^kg(w)}$$

exists almost everywhere and is finite on that set where the denominator for sufficiently large $n$ differs from zero, i.e. where at least one of the numbers $T^kg(w)>0$.

This theorem was formulated and proved by D.S. Ornstein and R.V. Chacon [CO] (see also [H], [N]); its analogue for continuous time has since been obtained (see [AC]).

Among the direct corollaries of the Ornstein–Chacon ergodic theorem are the Birkhoff ergodic theorem and various of its previously proposed generalizations, but there are also a number of ergodic theorems which are independent of the Ornstein–Chacon ergodic theorem, which is itself subject to various generalizations (see [C], [T], as well as the bibliography under Operator ergodic theorem). Of all the generalizations of the Birkhoff theorem, the most frequently used is the Ornstein–Chacon ergodic theorem.

Sometimes the Ornstein–Chacon ergodic theorem, as well as other theorems which deal with the limit of the ratio between two time-dependent means are called "ratio ergodic theorems" .

References

[CO]	R.V. Chacon, D.S. Ornstein, "A general ergodic theorem" Illinois J. Math. , 4 : 2 (1960) pp. 153–160
[H]	E. Hopf, "On the ergodic theorem for positive linear operators" J. Reine Angew. Math. , 205 (1960) pp. 101–106
[N]	J. Neveu, "Mathematical foundations of the calculus of probabilities" , Holden-Day (1965) (Translated from French)
[AC]	M.A. Alaoglu, J. Cunsolo, "An ergodic theorem for semigroups" Proc. Amer. Math. Soc. , 24 : 1 (1970) pp. 161–170
[C]	R.V. Chacon, "Convergence of operator averages" , Ergodic Theory (Proc. Internat. Symp. New Orleans, 1961) , Acad. Press (1963) pp. 89–120
[T]	T.R. Terrell, "A ratio ergodic theorem for operator semigroups" Boll. Un. Mat. Ital. , 6 : 2 (1972) pp. 175–180

Comments

In the Western literature one usually speaks of the Chacon–Ornstein ergodic theorem. For an overview of all kinds of ergodic theorems see [K]. An excellent account of the Chacon–Ornstein ergodic theorem is in [G].

References

[K]	U. Krengel, "Ergodic theorems" , de Gruyter (1985)
[G]	A. Garcia, "Topics in almost everywhere convergence" , Markham (1970)