On the limiting behaviour of geometric functionals of convex hulls of random walks

doctoral thesis

• Title (croatian): O graničnom ponašanju geometrijskih funkcionala konveksnih ljuski slučajnih šetnji
• Author: Tomislav Kralj
• Mentor: Nikola Sandrić (mentor)
• Mentor: Stjepan Šebek (komentor)
• Committee member: Bojan Basrak (predsjednik povjerenstva)
• Committee member: Hrvoje Planinić (član povjerenstva)
• Committee member: Stjepan Šebek (član povjerenstva)
• Committee member: Nikola Sandrić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2024-12-06, 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 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 \(L^2\) 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.

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 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 \(L^2\) 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.

• Keywords: random walk; central limit theorem; strong law of large numbers; convex hull; perimeter length; diameter
• Keywords (croatian): slučajna šetnja; centralni granični teorem; jaki zakon velikih brojeva; konveksna ljuska; duljina opsega; dijametar
• Language: english
• URN:NBN: urn:nbn:hr:217:058537
• Project: Number: 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 programme: Title: 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 resource: Text
• Extent: v, 136 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-01-31 10:25:16