prikaz prve stranice dokumenta Azumina nejednakost i primjene
Embargo period 2025-12-03
master's thesis
Azumina nejednakost i primjene
Zagreb: University of Zagreb, Faculty of Science, 2024. urn:nbn:hr:217:760838
Čabraja, Luka
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Čabraja, L. (2024). Azumina nejednakost i primjene (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:760838

Čabraja, Luka. "Azumina nejednakost i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2024. https://urn.nsk.hr/urn:nbn:hr:217:760838

Čabraja, Luka. "Azumina nejednakost i primjene." Master's thesis, University of Zagreb, Faculty of Science, 2024. https://urn.nsk.hr/urn:nbn:hr:217:760838

Čabraja, L. (2024). 'Azumina nejednakost i primjene', Master's thesis, University of Zagreb, Faculty of Science, accessed 11 January 2025, https://urn.nsk.hr/urn:nbn:hr:217:760838

Čabraja L. Azumina nejednakost i primjene [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2024 [cited 2025 January 11] Available at: https://urn.nsk.hr/urn:nbn:hr:217:760838

L. Čabraja, "Azumina nejednakost i primjene", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2024. Available at: https://urn.nsk.hr/urn:nbn:hr:217:760838

Metadata
TitleAzumina nejednakost i primjene
Title (english)Azuma's inequality and its applications
AuthorLuka Čabraja
MentorVjekoslav Kovač (mentor)
Committee memberVjekoslav Kovač (predsjednik povjerenstva)
Committee memberVanja Wagner (član povjerenstva)
Committee memberRudi Mrazović (član povjerenstva)
Committee memberBojan Basrak (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2024-11-27, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
U ovom radu bavili smo se Azuminom nejednakosti i njenim primjenama u kombinatorici. Sama Azumina nejednakost vezana je uz specifičnu vrstu slučajnih procesa, martingale. Uzmemo li sada martingal \(X\) čiji su prirasti ograničeni, tada Azumina nejednakost daje ogradu na vjerojatnost događaja da ćemo nakon \(n\) koraka otići "daleko" od očekivanja. U prvom poglavlju uveli smo teorijski pojam martingala i dali neke rezultate vezane uz njih. Nadalje, dokazali smo Azuminu nejednakost te usporedili nju i Černovljevu nejednakost s poznatom Markovljevom i Čebiševljevom ogradom. Također, obradili smo nekoliko poboljšanja glavne nejednakosti u nekim specifičnim slučajevima.\par Na kraju smo obradili neke primjene Azumine nejednakosti u kombinatorici. Primjeri su bili vezani uz slučajne grafove i kromatski broj, te takozvane pseudo-Booleove funkcije.
Abstract (english)
In this paper, we focused on Azuma's inequality and its applications in combinatorics. Azuma's inequality itself is related to a specific type of stochastic process, martingales. If we take a martingale \(X\) with bounded increments, Azuma's inequality provides an upper bound on the probability that, after \(n\) steps, we will be "far" from the expected value. In the first chapter, we introduced the theoretical concept of martingales and presented some results related to them. Furthermore, we proved Azuma's inequality and compared Azuma's and Chernoff's inequalities with the well-known Markov and Chebyshev bounds. We also explored several improvements to the main inequality in certain specific cases.\par Finally, we examined some applications of Azuma's inequality in combinatorics. All examples were related to random graphs, the chromatic number, and so-called pseudo-Boolean functions.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:760838
Study programmeTitle: Mathematical Statistics
Study programme type: university
Study level: graduate
Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
Type of resourceText
File originBorn digital
Access conditionsEmbargoed access
Embargo expiration date: 2025-12-03
Terms of useLicence Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) icon
Created on2024-11-12 15:15:36