TY - JOUR A1 - Frühwirth, Lorenz A1 - Prochno, Joscha T1 - Hölder’s inequality and its reverse — a probabilistic point of view JF - Mathematische Nachrichten N2 - In this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in the classical Hölder 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 Hölder ratio. KW - Berry–Esseen bound KW - central limit theorem KW - Hölder’s inequality KW - 𝓁 𝑛 𝑝 ball KW - large deviation principle KW - moderate deviation principle KW - reverse inequality Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:101:1-2023062315175042989607 VL - 296 IS - 12 SP - 5493 EP - 5512 PB - Wiley CY - Hoboken ER - TY - JOUR A1 - Aistleitner, Christoph A1 - Frühwirth, Lorenz A1 - Prochno, Joscha T1 - Diophantine conditions in the law of the iterated logarithm for lacunary systems JF - Probability Theory and Related Fields (ISSN: 1432-2064) N2 - 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 Erdős 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 Erdős–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. KW - Lacunary trigonometric sums KW - Law of the iterated logarithm KW - Diophantine equations Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:101:1-2406190937284.886045166092 SN - 0178-8051 SN - 1432-2064 VL - 192 IS - 1 SP - 545 EP - 574 PB - Springer CY - Berlin/Heidelberg ER - TY - THES A1 - Frühwirth, Lorenz T1 - The Asymptotic Behavior of Birkhoff- and Lacunary Sums N2 - 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. Y1 - 2025 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-15677 ER - TY - JOUR A1 - Frühwirth, Lorenz A1 - Juhos, Michael A1 - Prochno, Joscha T1 - The large deviation behavior of lacunary sums JF - Monatshefte für Mathematik N2 - 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. KW - Hadamard gap sequence KW - Large deviation principle KW - Large gap condition KW - Geometric progression Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:101:1-2022081723261050522140 VL - 2022 IS - 199 SP - 113 EP - 133 PB - Springer Nature CY - Berlin ER - TY - JOUR A1 - Frühwirth, Lorenz A1 - Prochno, Joscha T1 - Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls JF - Journal of Mathematical Analysis and Applications (Online ISSN: 1096-0813) N2 - 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 Poincaré-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 Poincaré-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V2-Orlicz radius. KW - Entropy maximization KW - Gibbs conditioning principle KW - Large deviation principle KW - Orlicz space KW - Poincaré-Maxwell-Borel lemma KW - Sanov's theorem Y1 - 2024 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus4-18723 VL - 2024 IS - 536,1 PB - Elsevier CY - Amsterdam ER -