New Proofs of the Basel Problem using Stochastic Processes
Uwe Hassler1, Mehdi H. Kouchack1, *
The variance of the standard Gumbel (or extreme value) distribution is equal to . This number also solves the so-called Basel problem which is equal to This latter problem was first solved by Leonhard Euler. A more detailed historical exposition is provided in Section 2.
Our first contribution is to classify the multitude of earlier proofs in Section 3. The second contribution consists of a new class of proofs.
Our method of proofs is rooted in the theory of stochastic processes. It relies on the Karhunen-Loève expansion of a Gaussian process and the related eigenstructure of the covariance kernel.
Using the cases of a (standard) Wiener process, of a demeaned or a detrended Wiener process, we provide new simple proofs of
We outlined the eigenstructure of different Gaussian processes that could be used to solve the Basel problem and give, as an example, the so-called Brownian bridge.
* Address correspondence to this author at Business Administratio and Economics, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 4, Frankfurt, Germany; Tel: 00496979834771;