Teorija velikih devijacija

master's thesis

• Author: Darko Dujmić
• Mentor: Ante Mimica (mentor)
• Committee member: Ante Mimica (predsjednik povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-07-11, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu, polazeći od predznanja teorije vjerojatnosti, bavili smo se vjerojatnostima rijetkih događaja, odnosno graničnim ponašanjem tih događaja. Dali smo osnovni uvod u teoriju velikih devijacija, ističući važnost Crámerovog teorema i Varadhanove leme. Kao uvod u temu, prezentirani su neki osnovni rezultati iz teorije vjerojatnosti i metričkih prostora. Detaljno smo dokazali Crámerov teorem, koji nam pokazuje kako izračunati funkciju intenziteta I. Pokazali smo da je funkcija intenziteta zapravo Legendre-Fanschelova transformacija funkcije \(\ln\varphi\). Bavili smo se i općom teorijom velikih devijacija, napuštajući jednako distribuirane slučajne varijable. Vidjeli smo da princip velikih devijacija slijedi iz Crámerovog teorema. Naveli smo jedan od najvažnijih rezultata teorije velikih devijacija, generalizaciju Laplaceove aproksimacije eksponencijalnih integrala, Varadhanovu lemu. Na kraju, pokazali smo da slaba verzija Stirlingove formule slijedi iz Varadhanove leme.

Abstract (english)

In this work, starting from foreknowledge of probability theory, we have dealt with probabilities of rare events, respectively with asymptotic estimates of rare events. We have given an elementary introduction to large deviations, highlighting the role of Crámer and Varadhan. As introduction to the main subject we have presented some elementary results from probability theory and metric spaces. We have proved Crámer’s theorem, which provides a general principle to compute the rate function I. We have shown that the rate function I is the Legendre-Fenchel transform of the function \(\ln\varphi\). We have developed a formal framework for the quantification of probabilities of rare events in which the complete theory of large deviations can be developed. We have shown that Crámer’s theorem is essentially large deviation principle. We have formulated important general theorem for large deviation theory, generalization of the Laplace method, Varadhan’s lemma. In the end we have shown that very weak version of Stirling’s formula follows from Varadhan’s lemma.

• Keywords: vjerojatnosti rijetkih događaja; Crámerov teorem; Varadhanova lema; Legendre-Fanschelova transformacija; opća teorija velikih devijacija
• Keywords (english): probabilities of rare events; Crámer’s theorem; Varadhan’s lemma; Legendre-Fenchel transform; complete theory of large deviations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:422509
• Study programme: Title: Mathematical Statistics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-01-15 14:28:32