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  <doc>
    <id>1628</id>
    <completedYear>2023</completedYear>
    <publishedYear>2023</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>5493</pageFirst>
    <pageLast>5512</pageLast>
    <pageNumber>20 Seiten</pageNumber>
    <edition/>
    <issue>12</issue>
    <volume>296</volume>
    <type>article</type>
    <publisherName>Wiley</publisherName>
    <publisherPlace>Hoboken</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2023-03-29</completedDate>
    <publishedDate>2023-03-29</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Hölder’s inequality and its reverse — a probabilistic point of view</title>
    <abstract language="eng">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.</abstract>
    <parentTitle language="deu">Mathematische Nachrichten</parentTitle>
    <identifier type="doi">10.1002/mana.202200411</identifier>
    <identifier type="urn">urn:nbn:de:101:1-2023062315175042989607</identifier>
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    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Lorenz Frühwirth</author>
    <author>Joscha Prochno</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Berry–Esseen bound</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>central limit theorem</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Hölder’s inequality</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>𝓁 𝑛 𝑝 ball</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>large deviation principle</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>moderate deviation principle</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>reverse inequality</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Fakultät für Informatik und Mathematik</collection>
    <collection role="Transformationsvertrag" number="">DEAL Wiley</collection>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/1628/Fruehwirth_Hoelders-inequality.pdf</file>
  </doc>
  <doc>
    <id>1665</id>
    <completedYear>2024</completedYear>
    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>545</pageFirst>
    <pageLast>574</pageLast>
    <pageNumber>30 Seiten</pageNumber>
    <edition/>
    <issue>1</issue>
    <volume>192</volume>
    <type>article</type>
    <publisherName>Springer</publisherName>
    <publisherPlace>Berlin/Heidelberg</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2024-04-03</completedDate>
    <publishedDate>2024-04-03</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Diophantine conditions in the law of the iterated logarithm for lacunary systems</title>
    <abstract language="eng">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.</abstract>
    <parentTitle language="eng">Probability Theory and Related Fields (ISSN: 1432-2064)</parentTitle>
    <identifier type="issn">0178-8051</identifier>
    <identifier type="issn">1432-2064</identifier>
    <identifier type="doi">10.1007/s00440-024-01272-6</identifier>
    <identifier type="urn">urn:nbn:de:101:1-2406190937284.886045166092</identifier>
    <enrichment key="opus.import.date">2025-07-29T13:00:52+00:00</enrichment>
    <enrichment key="opus.source">sword</enrichment>
    <enrichment key="opus.import.user">deepgreen</enrichment>
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    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Christoph Aistleitner</author>
    <author>Lorenz Frühwirth</author>
    <author>Joscha Prochno</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Lacunary trigonometric sums</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Law of the iterated logarithm</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Diophantine equations</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Fakultät für Informatik und Mathematik</collection>
    <collection role="Import" number="import">Import</collection>
    <collection role="Transformationsvertrag" number="">DEAL Springer Nature</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/1665/00440_2024_Article_1272.pdf</file>
  </doc>
  <doc>
    <id>1567</id>
    <completedYear>2025</completedYear>
    <publishedYear>2025</publishedYear>
    <thesisYearAccepted>2025</thesisYearAccepted>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>109 Seiten</pageNumber>
    <edition/>
    <issue/>
    <volume/>
    <type>doctoralthesis</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2025-04-14</completedDate>
    <publishedDate>2025-04-14</publishedDate>
    <thesisDateAccepted>2025-04-01</thesisDateAccepted>
    <title language="eng">The Asymptotic Behavior of Birkhoff- and Lacunary Sums</title>
    <abstract language="eng">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.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus4-15677</identifier>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Lorenz Frühwirth</author>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <thesisGrantor>Universität Passau</thesisGrantor>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/1567/Dissertation_Fruehwirt.pdf</file>
  </doc>
  <doc>
    <id>1542</id>
    <completedYear>2022</completedYear>
    <publishedYear>2022</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>113</pageFirst>
    <pageLast>133</pageLast>
    <pageNumber>21 Seiten</pageNumber>
    <edition/>
    <issue>199</issue>
    <volume>2022</volume>
    <type>article</type>
    <publisherName>Springer Nature</publisherName>
    <publisherPlace>Berlin</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2022-06-24</completedDate>
    <publishedDate>2022-06-24</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">The large deviation behavior of lacunary sums</title>
    <abstract language="eng">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.</abstract>
    <parentTitle language="deu">Monatshefte für Mathematik</parentTitle>
    <identifier type="doi">10.1007/s00605-022-01733-x</identifier>
    <identifier type="urn">urn:nbn:de:101:1-2022081723261050522140</identifier>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Lorenz Frühwirth</author>
    <author>Michael Juhos</author>
    <author>Joscha Prochno</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Hadamard gap sequence</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Large deviation principle</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Large gap condition</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Geometric progression</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Fakultät für Informatik und Mathematik</collection>
    <collection role="Transformationsvertrag" number="">DEAL Springer Nature</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/1542/Fruehwirth_LargeDeviationBehavior.pdf</file>
  </doc>
  <doc>
    <id>1872</id>
    <completedYear>2024</completedYear>
    <publishedYear>2024</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber>25 Seiten</pageNumber>
    <edition/>
    <issue>536,1</issue>
    <volume>2024</volume>
    <type>article</type>
    <publisherName>Elsevier</publisherName>
    <publisherPlace>Amsterdam</publisherPlace>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2024-01-30</completedDate>
    <publishedDate>2024-02-06</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls</title>
    <abstract language="eng">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.</abstract>
    <parentTitle language="eng">Journal of Mathematical Analysis and Applications (Online ISSN: 1096-0813)</parentTitle>
    <identifier type="doi">10.1016/j.jmaa.2024.128169</identifier>
    <identifier type="urn">urn:nbn:de:bvb:739-opus4-18723</identifier>
    <enrichment key="opus.doi.autoCreate">false</enrichment>
    <enrichment key="opus.urn.autoCreate">true</enrichment>
    <enrichment key="review.accepted_by">2</enrichment>
    <licence>Creative Commons - CC BY - Namensnennung 4.0 International</licence>
    <author>Lorenz Frühwirth</author>
    <author>Joscha Prochno</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Entropy maximization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Gibbs conditioning principle</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Large deviation principle</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Orlicz space</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Poincaré-Maxwell-Borel lemma</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Sanov's theorem</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Fakultät für Informatik und Mathematik</collection>
    <collection role="Transformationsvertrag" number="">DEAL Elsevier</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/1872/fruewirth_prochno_sanov-type.pdf</file>
  </doc>
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