@article{FruehwirthProchno2023, author = {Fr{\"u}hwirth, Lorenz and Prochno, Joscha}, title = {H{\"o}lder's inequality and its reverse β€” a probabilistic point of view}, series = {Mathematische Nachrichten}, volume = {296}, journal = {Mathematische Nachrichten}, number = {12}, publisher = {Wiley}, address = {Hoboken}, doi = {10.1002/mana.202200411}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2023062315175042989607}, pages = {5493 -- 5512}, year = {2023}, abstract = {In this article, we take a probabilistic look at H{\"o}lder's inequality, considering the ratio of terms in the classical H{\"o}lder inequality for random vectors in ℝ𝑛. We prove a central limit theorem for this ratio, which then allows us to reverse the inequality up to a multiplicative constant with high probability. The models of randomness include the uniform distribution on 𝓁𝑛𝑝 balls and spheres. We also provide a Berry-Esseen-type result and prove a large and a moderate deviation principle for the suitably normalized H{\"o}lder ratio.}, language = {en} } @article{AistleitnerFruehwirthProchno2024, author = {Aistleitner, Christoph and Fr{\"u}hwirth, Lorenz and Prochno, Joscha}, title = {Diophantine conditions in the law of the iterated logarithm for lacunary systems}, series = {Probability Theory and Related Fields (ISSN: 1432-2064)}, volume = {192}, journal = {Probability Theory and Related Fields (ISSN: 1432-2064)}, number = {1}, publisher = {Springer}, address = {Berlin/Heidelberg}, issn = {0178-8051}, doi = {10.1007/s00440-024-01272-6}, url = {http://nbn-resolving.de/urn:nbn:de:101:1-2406190937284.886045166092}, pages = {545 -- 574}, year = {2024}, abstract = {It is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by ErdΕ‘s and Fortet in the 1950s that probability theory's limit theorems may fail for lacunary sums (sum f(n_k x)) if the sequence ((n_k)_{k ge 1}) has a strong arithmetic "structure". The presence of such structure can be assessed in terms of the number of solutions k, l of two-term linear Diophantine equations (an_k - bn_l = c). As the first author proved with Berkes in 2010, saving an (arbitrarily small) unbounded factor for the number of solutions of such equations compared to the trivial upper bound, rules out pathological situations as in the ErdΕ‘s-Fortet example, and guarantees that (sum f(n_k x)) satisfies the central limit theorem (CLT) in a form which is in accordance with true independence. In contrast, as shown by the first author, for the law of the iterated logarithm (LIL) the Diophantine condition which suffices to ensure "truly independent" behavior requires saving this factor of logarithmic order. In the present paper we show that, rather surprisingly, saving such a logarithmic factor is actually the optimal condition in the LIL case. This result reveals the remarkable fact that the arithmetic condition required of ((n_k)_{k ge 1}) to ensure that (sum f(n_k x)) shows "truly random" behavior is a different one at the level of the CLT than it is at the level of the LIL: the LIL requires a stronger arithmetic condition than the CLT does.}, language = {en} } @phdthesis{Fruehwirth2025, author = {Fr{\"u}hwirth, Lorenz}, title = {The Asymptotic Behavior of Birkhoff- and Lacunary Sums}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-15677}, school = {Universit{\"a}t Passau}, pages = {109 Seiten}, year = {2025}, abstract = {This doctoral thesis consists of three independently published research articles on the asymptoic behaviour of Lacunary- and Birkhoff sums. The former are sums formed by periodic functions and exponentially growing sequences of natural numbers. The corresponding summands often exhibit behavior typical of independent and identically distributed random variables. The methods used are of an analytical and probabilistic nature. The Birkhoff sums considered in this work are generated by the Kronecker sequence and by discontinuous functions. The methods employed are from the field of metric number theory, specifically classical results from continued fraction theory are utilized.}, 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 :=βˆ‘[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 β†’ |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} } @article{FruehwirthProchno2024, author = {Fr{\"u}hwirth, Lorenz and Prochno, Joscha}, title = {Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls}, series = {Journal of Mathematical Analysis and Applications (Online ISSN: 1096-0813)}, volume = {2024}, journal = {Journal of Mathematical Analysis and Applications (Online ISSN: 1096-0813)}, number = {536,1}, publisher = {Elsevier}, address = {Amsterdam}, doi = {10.1016/j.jmaa.2024.128169}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus4-18723}, pages = {25 Seiten}, year = {2024}, abstract = {In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-2 large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincar{\Β΄e}-Maxwell-Borel lemma, we deduce a conditional limit theorem for high-dimensional Orlicz balls. In more geometric parlance, the latter shows that if V1 and V2 are Orlicz functions, then random points in the V1-Orlicz ball, conditioned on having a small V2-Orlicz radius, look like an appropriately scaled V2-Orlicz ball. In fact, we show that the limiting distribution in our Poincar{\Β΄e}-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V2-Orlicz radius.}, language = {en} }