Ovaj diplomski rad ima tri poglavlja. U prvome smo definirali Brownovo gibanje koje nam je vrlo bitno za cjelokupni rad. Pojam slabe konvergencije u metričkom prostoru se proteže kroz rad, a u prvom poglavlju smo pokazali neke osnovne rezultate za tu vrstu konvergencije. Definirali smo i konvergenciju po distribuciji i napetost. Prvo poglavlje nam je poslužilo za uvod u problematiku rada i dalo nam osnovne alate za rad u prostorima C i D. U drugom poglavlju promatrali smo prostor neprekidnih funkcija na intervalu [0, 1], prostor C. Nakon Prohovljevog teorema koji nam je dao vezu između relativne kompaktnosti i napetosti familije vjerojatnosnih mjera, promatrali smo par teorema koji su nam dali uvjete za slabu konvergenciju i napetost. Spomenuli smo i Wienerovu mjeru na tom prostoru, te smo dokazali Donskerov teorem u tom okruženju. Za dokaz Donskerovog teorema koristili smo rezultate iz centralnih graničnih teorema, kao i puno rezultata koji su prije navedeni u tekstu. U trećem poglavlju promatramo prostor cádlág funkcija, D, prostor funkcija koje su zdesna neprekidne i imaju lijevi limes. Na tom prostoru smo promatrali Skorohodovu topologiju i uočili razlike između prostora D i prostora C. Našli smo novu metriku za prostor D, različitu od one za C, no D nije bio potpun pod tom metrikom pa smo ju morali modificirati i na kraju smo dobili metriku pod kojom je D potpun. Pomoću novih modula neprekidnosti, dobili smo i nove iskaze teorema. Ipak, dosta rezultata iz prostora C vrijedi i u prostoru D, uz male preinake. Prije dokaza Donskerovog teorema naveli smo kriterij za konvergenciju kojega smo koristili pri samom dokazu. Dokazali smo Donskerov teorem na dva načina, jedan vrlo sličan dokazu u prostoru C, dok je drugi dokaz bio nov.

This master thesis has three chapters. In first we have defined Brownian motion that is very important for the whole thesis. Definition of weak convergence in metric space was necessary for our results and in first chapter we have shown some basic results for that type of convergence. As well, we have defined convergence in distribution and tightness. Purpose of first chapter was to give us basic tools for usage in spaces C and D In the second chapter, we have looked at the space of continuous functions on the interval [0, 1], space C. After Prohorov’s theorem, which gave us connection between relative compactness and tightness of family of probability measures, we have looked at few theorems that gave us conditions for weak convergence and tightness. We have mentioned Wiener measure on that space, and proved Donsker’s theorem. For that proof we have used results from central limit theorems, together with results mentioned before in the text. In the third chapter we have looked at space of cádlág functions, ´ D, functions that have left-hand limit and are right continuous. At that space we have looked at Skorohod topology and seen differences between space D and space C. We have found a new metric for space D, but D was not complete under that metric so we needed to modify that metric a bit in order to finally get metric under which D is complete. Using new modulus of continuity, we got different theorems. Although, a lot of results for C were applicable to D, maybe with small changes. Before the proof of Donsker’s theorem, we have mentioned one convergence criterion that was used for proof. We have proven Donsker’s theorem in two ways, one very similar to proof for space C, and the other one was something new.