In the last week of May I attended two lectures given by Professor Matthew Nicol.
Let be a prob space with a -algebra . Let be a sub -algebra.
Example. on , and be the Borel -algebra. Let . Note that .
Let be a -measurable r.v. and . The conditional expectation is the unique -measurable r.v. satisfying for all , and for all .
Note that if and only if is -measurable; and if is independent of .
Let be a stationary ergodic process with stationary initial distribution . A basic problem is to find sufficient conditions on and on functions such that satisfies the central limit theorem (CLT) , where the limit variance is given by .
Let be a conservative diffeomorphism on . There are two operators: , and via for all test function .
Property. (vol-preserving) and .
Let be an increasing sequence of -algebras. Then a sequence of r.v. is called a martingale w.r.t. , if is -measurable, .
Let be a decreasing sequence of -algebras. Then a sequence of r.v. is called a reverse martingale w.r.t. , if is -measurable, for any .
Theorem. Let be a stationary ergodic sequence of (reverse) martingale differences w.r.t. . Suppose , and . Then in distribution.
Gordin: Suppose is ergodic. Consider the Birkhoff sum for some with . The time series can be approximated by martingale differences provided the correlations decay quickly enough.
Suppose there exists with , such that . Then define , and let .
Property. Let be such that is decreasing. is a reverse martingale with respect to .
Proof. Note that . Then .
Let . It remains to show . To this end, we pick an element and write it as for some . Then . This completes the proof.
Three theorems of Gordin. Let be an invertible -preserving ergodic system, and be a strictly stationary ergodic sequence.
Theorem 1. Suppose there exists such that , . Then (*) implies exists, and in distribution (degenerate if ).
–Mixing condition. Let .
Theorem 2. Suppose for some , and . Then (*) implies the conclusion of Theorem 1.
–uniform mixing condition. Let .
Theorem 3. Suppose and . Then (*) implies the conclusion of Theorem 1.
Cuny–Merlevede: not only the CLT, but also the ASIP holds under the above conditions.
Note that we started with an invariant measure . The operator and can be defined for all non-conservative maps. To emphasize the difference, we use . Suppose for some . Then is an absolutely continuous invariant prob. measure:
Then we can rewrite , in the sense that