Seminar za teoriju vjerojatnosti /
Probability Seminar
Department of Mathematics
University of Zagreb
Croatia |
2013./2014.:
Najavljena predavanja:
Generalizacija martingalne transformacije i njena primjena na
paraprodukte i stohasticke integrale
Simulation of Stochastic Processes
Abstract: This Erasmus-mini-course addresses students who have a basic knowledge in probability theory.
In this course you will learn
- basic techniques to generate samples of univariate distributions
- how to simulate stochastic processes
- the Monte Carlo Method
The focus will be on the normal distribution and Brownian motion.
References:
B. Boettcher, Simulation of Brownian Motion,
Chapter in the
Textbook: Brownian Motion - An Introduction to Stochastic Processes, R.
Schilling, L. Partzsch, De Gruyter, 2012.
Odrzana predavanja:
Fluktuacije turbulentnog fluida vodjenog difuzijama sa
skokovima
Kvantitativni rezultati konvergencije ergodickih usrednjenja
(Quantitative results on the convergence of ergodic averages)
Apsolutna neprekidnost i singularnost vjerojatnosnih mjera
induciranih cisto prekidnom Girsanovljevom transformacijom izotropnog
stabilnog procesa
(Absolute continuity and singularity of probability measures induced by a purely discontinuous Girsanov transform of a stable process)
Granicno ponasanje slucajnih varijabli opazenih u slucajnom vremenu
(obrana doktorske disertacije)
(Limiting behaviour of random variables observed in random times)
Stohasticka preferencija i medicinska dijagnoza
Sazetak: Na predavanju ce biti govora o stohastickoj preferenciji i poopcenju
teorema reprezentacije koji govori da konzistentnost (preferencije)
povlaci njenu numericku reprezentaciju koja je vjerojatnost. Poopcenja
se odnose na nekonzistentne stohasticke preferencije. Zatim ce biti
govora o izvjesnom
poopcenju Bayesove formule i agregaciji prioriteta s primjenom na
postavljanje medicinske dijagnoze. Na kraju će biti primjer jedne dijagnoze
sa zavisnim simptomima. Ukratko ce biti opisana metoda potencijala koja je
u pozadini spomenutih poopcenja.
(Abstract: In this lecture we will speak about the stochactic preference and a
generalization of the reprezentation theorem for the inconsistent
preference.
We will give a short description of the potential method and two applications
on the diagnosis in medicine. The first one concerns Bayes type
aggregation of priorities, the second one concerns a diagnosis with
dependent symptoms.)
Vjerojatnost propasti i distribucija supremuma za generalizirane procese rizika IV
(Ruin probability and distribution of supremum for generalized risk processes IV)
Central Limit Theorems for Supercritical Branching Markov Processes
Abstract: In this talk I will present
some recently established spatial central limit theorems for a large class
of supercritical branching Markov processes with general spatial-dependent
branching mechanisms. These results are generalizations of the spatial
central limit theorems proved in Adamczak and Milos (2011) for branching OU
processes with binary branching mechanisms. Compared with the results of
Adamczak and Milos (2011), our central limit theorems are more satisfactory
in the sense that the normal random variables in our theorems are
non-degenerate.
The talk is based on joint work with Y.-X. Ren and R. Zhang.
Ekstremalna svojstva slabo zavisnih vremenskih i prostornih nizova III
18:00
Stohasticka jednadzba provodjenja kao limes Brownovog gibanja
ideksiranog rombovskom mrezom
(Stochastic heat equation as a limit of a Brownian motion indexed by a
rhombus graph)
Sazetak: Na pocetku dat cemo kratki pregled trenutnih istrazivanja u podrucju
slucajnih procesa indeksiranih vremenolikim grafovima
(geometrijskim problemima, interpretaciji, vezama s granajucim
procesima, skrivenim markovljevim modelom).
Glavni dio predavanja odnosit ce se na vezu ovih procesa s
stohastickom jednadzbom provodjenja.
Kod slucajnih procesa indeksiranih grafovima postavlja se prirodno
pitanje - sto se dogadja kad reprezentacija grafa postaje gusta na nekom
podskupu ravnine.
Prvo cemo se baviti problemom 'dobre' definicije ovog procesa,
i zasto je slutnja Burdzy-ja i Pal-a iz 2011 tocna.
Nakon toga sto se dogadja kad imamo dvostrano Brownovo gibanje
na rombovskoj mrezi, a dijagonale rombova konvergiraju prema nuli
zadanim brzinama. Pokazat ce se da u granicnom slucaju dobivamo
stohasticku jednadzbu provodjenja.
(Abstract: The plane is tiled by cells of specific form that make up a graph, and on the (representation of) edges of this graph we define a process. The question is what happens to the process when the cell size goes to zero? Is there a process in the limit? What will that process (indexed by the whole plane) be?
Burdzy and Pal in their paper studied such a process on a honeycomb graph for a Brownian motion type process, and got that the covariance structure is non-trivial in the limit.
We will talk about a rhombus graph for the same process, and show what happens in two cases when the size of the cell goes to zero. We will also comment how to interpret of the results.)
Ekstremalna svojstva slabo zavisnih vremenskih i prostornih nizova II
Povratnost i prolaznost procesa Levyjevog tipa
(Recurrence and transience of Levy-type processes)
Sazetak: U ovom seminaru ce biti predstavljeni uvjeti Chung-Fuchsovog tipa za
povratnost i prolaznost klase Fellerovih procesa vezane za
pseudodiferencijalne operatore. Kao posljedicu tih rezultata, diskutirat
cemo povratnost i prolaznost Fellerovih procesa obzirom na dimenziju
prostora stanja i Pruittove indekse i povratnost i prolaznost
Feller-Dynkinovih difuzija i procesa stabilnog tipa (stable-like
processes). Nadalje, u jednodimenzionalnom simetricnom slucaju, diskutirat
cemo perturbacije Fellerovih procesa koje nece uzrokovati promjenu
svojstva povratnosti i prolaznosti samog procesa, te cemo komentirati
dovoljne uvjete za povratnost i prolaznost u terminima pripadne Levyjeve
mjere. Konacno, komentirat cemo i neke uvjete usporedivanja za povratnost
i prolaznost takoder u terminima pripadnih Levyjevih mjera.\
(Abstract: We present Chung-Fuchs type conditions for the recurrence and transience
of Feller processes associated with pseudo-differential operators.
Further, we also discuss the recurrence and transience of Feller processes
with respect to the dimension of the state space and Pruitt indices and
the recurrence and transience of Feller-Dynkin diffusions and stable-like
processes (in the sense of R. Bass). Finally, in the one-dimensional
symmetric case, we discuss perturbations of Feller processes which do not
affect their
recurrence and transience properties, and we discuss sufficient conditions
for their recurrence and transience in terms of the corresponding Levy
measure. In addition, some comparison conditions for recurrence and
transience also in terms of the Levy measures are obtained.)
O Soricevoj metodi
(On Soric's method)
Sazetak:
Seminar je posvecen proslavljenom radu dr. Branka Sorica o
problemu visestrukog testiranja hipoteza i tzv. FDR metodi.
Rad je u cetvrt stoljeca od objavljivanja postao jedan od najsire
prihvacenih znanstvenih doprinosa u statistici. Na seminaru
otkrivamo do sada nepoznate veze i odnose Soricevog rada s
drugim, danas vrlo popularnim statistickim metodama.
On approximation with redundant dictionaries
Abstract:
Data approximation using sparse linear expansions from overcomplete
dictionaries has become a central theme in signal and image processing
with applications ranging from data acquisition (compressed sensing)
to denoising and compression.
For a given dictionary, we can also study best m-term approximation
rates for any specific function. Interestingly, the notions of sparse
expansions and certain asymptotic approximation rates are closely
linked in the case of nice non-redundant dictionaries (e.g., an
orthonormal basis in a Hilbert space.)
In this talk, I will explore the link between sparse expansions from
an overcomplete dictionary and asymptotic approximation rates.
Redundancy complicates the analysis, and we show that the close link
between the two notions fails in general.
However, using a probabilistic approach, we show that the close link
is retained for 'many' redundant dictionaries.
Stabilnost Schauderovih baza
(Stability of Schauder bases)
Sazetak: U ovom zajednickom radu s M.Nielsenom (Aalborg University)
povezujemo klasu uvjetnih Schauderovih baza sa Garnett-Jonesovom
udaljenosti u BMO prostorima.
Estimates on the transition density
of Markov process
Prosli seminari / Past seminars: