Probabilistic Methods in Telecommunications by Benedikt Jahnel & Wolfgang König

Author:Benedikt Jahnel & Wolfgang König
Language: eng
Format: epub
ISBN: 9783030360900
Publisher: Springer International Publishing

(6.4.3)

The -convergence in (6.4.3) is called the mean ergodic theorem , while the almost sure convergence is called the individual ergodic theorem . Wiener’s ergodic theorem is an extension of Birkhoff’s theorem in the form (6.3.2) in Remark 6.3.4; we took a test function of the entire point process.

Proof

We follow the presentation of the original proof of [Wie39] that is given in [Geo11, Chapter 14.A]. The strategy is the following. We first use Hilbert space arguments and convexity to show, for , the convergence in -sense, more precisely, first the convergence of the norms, then the -convergence. Then we extend the convergence to -convergence for , using that is dense in . As a preparation for the proof of the almost-sure convergence, one proves the maximal ergodic lemma, which controls the probability that the supremum over n of exceeds a constant, i.e., a tightness result. The final proof of the almost-sure convergence applies the Boreal–Cancelli lemma and is rather technical. Here is the first step:

Lemma 6.4.2(Minimizers of Convex Sets in Hilbert Spaces) Let be a nonempty closed convex set in a Hilbert space . Then there is a unique such that . Furthermore, any asymptotically minimizing sequence in converges towards f.

Proof of Lemma 6.4.2 Put and pick some sequence in such that limn→∞∥f n∥ = c. In any Hilbert space, the parallelogram identity

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