@phdthesis{Juhos2024, author = {Juhos, Michael}, title = {Probabilistic and geometric aspects of classical and non-commutative lp-type spaces in high dimensions}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-14857}, school = {Universit{\"a}t Passau}, pages = {viii, 157 Seiten}, year = {2024}, abstract = {This cumulative dissertation contains selected contributions to the field of asymptotic geometric analysis and high-dimensional probability. It is divided into two chapters: Chapter 1 explains some of the necessary theoretical background. In Section 1.1 it first gives a very concise history of asymptotic geometric analysis in general and then of the objects under study in particular, setting out some cornerstones in the discovery of the functional-analytic, geometric, and probabilistic properties of the spaces under consideration. The next section (1.2) gives the precise definitions and very basic properties of the three lp-type spaces that play a role in the contributed articles: the classical lp-sequence spaces, the mixed-norm sequence spaces, and the Schatten-classes Sp, each in its infinite- and finite-dimensional version. Section 1.3 is dedicated to the interplay between geometry and probability, expounding the general idea, introducing a few of the common tools, and exemplifying these on two kinds of limit theorems: Schechtman-Schmuckenschl{\"a}ger-type results and Poincar{\´e}-Maxwell-Borel lemmas. The first chapter concludes with Section 1.4, addressing a small sample of open questions pertaining to the contributed articles which are not answered in said articles and may be the interest of future research. The entirety of Chapter 2 consists of the contributed articles.}, language = {en} } @article{FruehwirthJuhosProchno2022, author = {Fr{\"u}hwirth, Lorenz and Juhos, Michael and Prochno, Joscha}, title = {The large deviation behavior of lacunary sums}, series = {Monatshefte f{\"u}r Mathematik}, volume = {2022}, journal = {Monatshefte f{\"u}r Mathematik}, number = {199}, publisher = {Springer Nature}, address = {Berlin}, doi = {10.1007/s00605-022-01733-x}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2022081723261050522140}, pages = {113 -- 133}, year = {2022}, abstract = {We study the large deviation behavior of lacunary sums (Sn /n)n∈N with Sn :=∑[k=1...n] f (a(k)U), n ∈ |N, where U is uniformly distributed on [0, 1], (a(k))k∈|N is an Hadamard gap sequence, and f : |R → |R is a 1-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed n and with a good rate function which is the same as in the case of independent and identically distributed random variables U(k), k ∈ |N, having uniform distribution on [0, 1]. When the lacunary sequence (a(k))k∈|N is a geometric progression, then we also obtain large deviation principles at speed n, but with a good rate function that is different from the independent case, its form depending in a subtle way on the interplay between the function f and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, preprint, 2020] who initiated this line of research for the case of lacunary trigonometric sums.}, language = {en} }