In this paper, we study the discretization of ergodic Functional Central Limit Theorem (CLT) established by Bhattacharya (see in Z Wahrscheinlichkeitstheorie Verwandte Geb 60:185–201, 1982) which states following: Given a stationary and Markov process $$(X_t)_{t \geqslant 0}$$ with unique invariant measure $$\nu $$ infinitesimal generator A, then, for every smooth enough function f, $$(n^{1/2} ...

A remarkable theorem of Besicovitch is that an integrable function f f on alttext="double-struck upper R squared"> <mml:mrow class...

We present generalizations of the classical Birkhoff and von Neumann ergodic theorems, where the time average is replaced by a more general average, including some density.

In this note we intend to present basic ergodic theory. We begin with the notion of a measure preserving transformation. We then define ergodicity and provide examples. Finally we sketch Birkhoff’s Ergodic Theorem and elucidate it with some examples.

(n) (x) · v). The map F is called a linear cocycle. Fix some norm ·· on R d , and use the same symbol to indicate the induced operator norm.

Partial results were obtained previously by K. F. Roth (1952) who established the existence of arithmetic progressions of length three in subsets of Z of positive upper density, and by E. Szemerédi (1969) who proved the existence of progressions of length four. In 1976 Furstenberg noticed that the statement of Theorem I is equivalent to a statement about "multiple recurrence" of measure-preserv...