The Individual Ergodic Theorem of Information Theory
نویسندگان
چکیده
منابع مشابه
The Converse of the Individual Ergodic Theorem
converge almost everywhere to a finite limit f*(x). It then follows that the limit function/* is integrable and that/*(7x) =/*(x) almost everywhere. This result can be applied to certain cases in which the given measure m is not preserved by the transformation T. In order to discuss this application, we recall some terminology for measures and transformations. If (X, S) is a measurable space, a...
متن کاملStrongly Ergodic Sequences of Integers and the Individual Ergodic Theorem
Let S = {ki,ki, ...} be an increasing sequence of positive integers. We call S strongly ergodic if for every measure preserving transformation T on a probability space (Cl, J, P) and every / £ Li(f2) we have limn-»oo(l/n) J^^j f(TkiuJ) = Pf(w) a.e. where Pf is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence S to be strongly ergodi...
متن کاملErgodic theorem, ergodic theory, and statistical mechanics.
This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundam...
متن کاملSzemerédi’s Theorem via Ergodic Theory
This essay investigates Furstenberg’s proof of Szemerédi’s Theorem. The necessary concepts and results from ergodic theory are introduced, including the Poincaré and Mean Ergodic Theorems which are proved in full. The Ergodic Decomposition Theorem is also discussed. Furstenberg’s Multiple Recurrence Theorem is then stated and shown to imply Szemerédi’s Theorem. The remainder of the essay concen...
متن کاملIndividual ergodic theorem for intuitionistic fuzzy observables using intuitionistic fuzzy state
The classical ergodic theory hasbeen built on σ-algebras. Later the Individual ergodictheorem was studied on more general structures like MV-algebrasand quantum structures. The aim of this paper is to formulate theIndividual ergodic theorem for intuitionistic fuzzy observablesusing m-almost everywhere convergence, where m...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Annals of Mathematical Statistics
سال: 1957
ISSN: 0003-4851
DOI: 10.1214/aoms/1177706899