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doctoral thesis
On the limiting behaviour of geometric functionals of convex hulls of random walks
Zagreb: University of Zagreb, Faculty of Science, 2024. urn:nbn:hr:217:058537
Kralj, Tomislav
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Kralj, T. (2024). On the limiting behaviour of geometric functionals of convex hulls of random walks (Doctoral thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:058537

Kralj, Tomislav. "On the limiting behaviour of geometric functionals of convex hulls of random walks." Doctoral thesis, University of Zagreb, Faculty of Science, 2024. https://urn.nsk.hr/urn:nbn:hr:217:058537

Kralj, Tomislav. "On the limiting behaviour of geometric functionals of convex hulls of random walks." Doctoral thesis, University of Zagreb, Faculty of Science, 2024. https://urn.nsk.hr/urn:nbn:hr:217:058537

Kralj, T. (2024). 'On the limiting behaviour of geometric functionals of convex hulls of random walks', Doctoral thesis, University of Zagreb, Faculty of Science, accessed 28 March 2025, https://urn.nsk.hr/urn:nbn:hr:217:058537

Kralj T. On the limiting behaviour of geometric functionals of convex hulls of random walks [Doctoral thesis]. Zagreb: University of Zagreb, Faculty of Science; 2024 [cited 2025 March 28] Available at: https://urn.nsk.hr/urn:nbn:hr:217:058537

T. Kralj, "On the limiting behaviour of geometric functionals of convex hulls of random walks", Doctoral thesis, University of Zagreb, Faculty of Science, Zagreb, 2024. Available at: https://urn.nsk.hr/urn:nbn:hr:217:058537

Metadata
TitleOn the limiting behaviour of geometric functionals of convex hulls of random walks
Title (croatian)O graničnom ponašanju geometrijskih funkcionala konveksnih ljuski slučajnih šetnji
AuthorTomislav Kralj
MentorNikola Sandrić (mentor)
MentorStjepan Šebek (komentor)
Committee memberBojan Basrak (predsjednik povjerenstva)
Committee memberHrvoje Planinić (član povjerenstva)
Committee memberStjepan Šebek (član povjerenstva)
Committee memberNikola Sandrić (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2024-12-06, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Universal decimal classification
(UDC)		51 - Mathematics
Abstract
In this dissertation, we study the asymptotic behavior of the convex hull generated by several mutually independent random walks. In the first chapter, we show that the convex hull, appropriately scaled, almost surely converges to the convex hull generated by the corresponding drift vectors and the origin. Then, by applying the continuous mapping theorem, we also demonstrate almost sure convergence of all intrinsic volumes, which implies almost sure convergence of the perimeter. Using a similar... More argument, we obtain almost sure convergence of the diameter process. In the next chapter, we move on to explore the distributional limit of the perimeter process. To successfully control the variance, we use the technique of martingale difference sequences and Cauchy’s formula. In the end, we obtain a very interesting and intuitive L2 approximation for the deviation of the perimeter process. Under certain assumptions about the drift vectors of the random walks, we determine the asymptotic behavior of the variance of the perimeter. We can establish a normal distributional limit if this asymptotic variance is positive. Following this, we focus on the diameter process, where it becomes crucial to note that the mapping that assigns diametral segments to polygons with the unique diametral segment is continuous. Finally, with the additional assumption about the set of drift vectors, we achieve results analogous to those for the perimeter process. In the final chapter, we will study the convex hull of centroids generated by a single random walk. We open a discussion on generalizing the assumptions made for the observed random walks and provide a detailed simulation study to explore what happens when these assumptions are not satisfied. Less
Abstract (croatian)
U ovoj disertaciji istražujemo granično ponašanje konveksne ljuske definirane pomoću nezavisnih slučajnih šetnji. U prvom poglavlju dokazujemo da konveksna ljuska, prikladno skalirana, gotovo sigurno konvergira prema konveksnoj ljusci definiranoj pripadnim drift vektorima i nulom. Primjenom teorema o neprekidnom preslikavanju, dokazujemo gotovo sigurnu konvergenciju svih intrinzičnih volumena, što implicira gotovo sigurnu konvergenciju opsega. Korištenjem slične argumentacije postiže se gotovo... More sigurna konvergencija procesa dijametra. U idućem poglavlju analiziramo distribucijski limes procesa opsega. Kako bismo uspješno kontrolirali varijancu, koristimo tehniku niza martingalnih razlika i Cauchyjevu formulu. Na kraju, dobivamo izuzetno zanimljivu i intuitivnu L2 aproksimaciju za devijaciju procesa opsega. Pod određenim pretpostavkama o drift vektorima slučajnih šetnji, određujemo granično ponašanje varijance opsega, te ako je ta granična varijanca pozitivna, možemo zaključiti normalni distribucijski limes. Nakon toga, usmjeravamo se na proces dijametra, gdje se uz prethodno spomenute alate ključnom pokazuje činjenica da je preslikavanje koje poligonima pridružuje dijametralne segmente neprekidno. Na kraju, uz dodatnu pretpostavku o skupu drift vektora, postižu se analogni rezultati kao i za proces opsega. U završnom poglavlju, promatramo konveksnu ljusku generiranu s centrima mase jedne planarne slučajne šetnje. Nakon toga, otvaramo diskusiju o poopćavanju pretpostavki koje smo postavili za promatrane slučajne šetnje i pružamo simulacijsku studiju koja istražuje posljedice napuštanja tih pretpostavki. Time se osvjetljavaju otvoreni problemi koji proizlaze iz ove disertacije. Less
Keywords
Keywords (croatian)
Languageenglish
URN:NBNurn:nbn:hr:217:058537
ProjectNumber: IP-2022-10-2277
Title: Geometrijska svojstva i teorija potencijala slučajnih procesa
Title: Geometric properties and potential theory of stochastic processes
Acronym: GPPTSP
Leader: Nikola Sandrić
Jurisdiction: Croatia
Funder: Hrvatska zaklada za znanost
Funding stream: Research Projects
Study programmeTitle: Doctoral study
Study programme type: university
Study level: postgraduate
Academic / professional title: doktor/doktorica u području prirodnih znanosti (doktor/doktorica u području prirodnih znanosti)
Type of resourceText
Extentv, 136 str.
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2025-01-31 10:25:16
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