On some problems of the modern probability theory  Page description

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Details of project

  
Identifier
37886
Type K
Principal investigator Major, Péter
Title in Hungarian A modern valószínűségszámítás néhány kérdéséről
Title in English On some problems of the modern probability theory
Panel Mathematics and Computing Science
Department or equivalent Alfréd Rényi Institute of Mathematics
Participants Berkes, István
Csáki, Endre
Révész, Pál
Starting date 2002-01-01
Closing date 2006-12-31
Funding (in million HUF) 11.634
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
Kutatásaink 4 témakörből álltak. Arch és Garch folyamatok és általánosításuk Ezek a pénzügyi matematikában fontos modellek bonyolult lineáris idősorok. Határeloszlástételeket és statisztikai eredményeket bizonyítottunk rájuk. Wiener folyamatok Ezek lokális idejével és a közönséges bolyongás ehhez kapcsolódó problémáival foglalkoztunk. Megadtuk a lokális idő Hilbert transzformáltjának és Cauchy-féle főértékének viselkedését leíró valószínűségi törvényeket, a Wiener excursion és a Bahadur-Kiefer folyamat legfontosabb tulajdonságait. Wiener folyamat lokális és magas dimenziós bolyongások tartózkodási ideje között szoros a kapcsolat. Itt Erdős és Taylor eredményeit javítottuk. Több erős beágyazási tételt bizonyítottunk. Véletlen integrálok Vettük egy normált empirikus mérték önmagával vett direkt szorzatát. Egy többváltozós függvény eszerinti integráljának es ilyen integrálok szuprémumának eloszlására adtunk éles becslést. Ehhez több távoli matematikai elméletet kellett alkalmaznunk. A bizonyított eredmények lehetővé teszik fontos statisztikai módszerek általánosítását. Megmagyarázzák, hogy lehet normált empirikus eloszlásfüggvény funkcionáljait Gauss folyamatok funkcionáljaival közelíteni, és hol vannak e közelítés határai. Véletlen törvényeket teljesítő számelméleti függvények Bebizonyítottuk az iterált logaritmus tétel élesítését és megmutattuk, hogy n_k\alpha alakú számsorozatok diszkrepanciái az n_k sorozat számelméleti tulajdonságaitól függő véletlen törvényeket teljesítenek.
Results in English
Our research consists of 4 subjects. Arch and Garch process, their generalizations This is an important model in financial mathematics. They are hard non-linear time series. We proved limit theorems and useful statistical results for them. Wiener processes We dealt with their local time and some occupation time problems of random walks. We gave the probabilistic laws of the Hilbert transform and the Cauchy principle value of their local time. We described the most important properties of the Wiener excursion and Bahadur-Kiefer process. The local time of the Wiener process and occupation time of high dimensional random walk are closely related. In this field we improved the results of Erdos and Taylor. We also proved strong embedding results. Multiple random integrals We took the direct product of a normed empirical distribution with itself. We gave sharp bounds on the integral of a function of several variables with respect to it and on the distribution of the supremum of such integrals. We applied several different mathematical theories in the proofs. Our results make possible to generalize some useful statistical methods. They explain how the functionals of normed empirical distributions can be approximated by Gaussian ones, and where the bounds of such approximations are. Number theoretic functions satisfying probabilistic laws We proved refinements of the law of iterated logarithm and showed that the discrepancies of a series of numbers n_k\alpha satisfy probabilistic laws depending on the diophantine properties of the series n_k.
Full text http://real.mtak.hu/391/
Decision
Yes





 

List of publications

  
Berkes, I., Horvath. L., Kokoszka. P., Shao, Q. M.: Almost sure convergence of the Bartlett estimator, Periodica Mathematica Hungarica 51: 11-25, 2005
Berkes, I., Gombay E., Horvath L. Kokoszka, P.:: Sequential change-point detection in Garch (p,q) maodels, Econometric Theory 20 1140-1167, 2004
Berkes I. ,Horvath L., Huskova, M., Steinebach J.: Application of permutations to the simulation of critical values., Nonparametric Stat.. 16 197-216, 2004
Berkes I., Horvath L., Kokoszka, P.: Probabilistictic and statistical properties of Garch processes., Asymptotic Results in Stochastics, Fields Institute Communications 44 409-429, 2004
Berkes I., Horvath L., Kokoszka, P.: Testing for parameter constancy in GARCH (p,q) models., Stat. Probab. Letters 70 1140-1167, 2004
Berkes, I., Weber, M.: Upper and lower class tests along subsequences., Stoch. Proc. Appl. 115: 679-700, 2005
Berkes I., Horvath L., Kokoszka P.: Near integrated Garch sequences., Annals of Applied Probability 15 890-913, 2005
Berkes I., Horvath L., Kokoszka P., Shao, Q. M.: On discriminating between long range dependence in the mean., Annals of Statistics,34 1140 -1165, 2006
Berkes, I., Horvath. L., Kokoszka. P.: A weighted goodness-of fit test for Garch (1.1) specification., Lithunaian Math. Journal 44 409-429, 2004
Csaki E., Revesz P., Shi Z.: Large void zones and occupation times for coalescing random walks., Stochastic Processes and their Applications 111, 97-108, 2004
Csaki E., Hu Y.,: On ranked excursion height of a Kiefer process., Journal of Theoretical Probability 17, 145-163, 2004
Csaki E., Foldes A., Shi Z.:: Our joint work with Miklos Csorgo, Asymptotic Methods in Stochastics: Festschrift for Miklos Csorgo, Fields Institute Communications 44 1-22, 2004
Csaki E., Hu, Y.: Invariance principles for ranked excursion lengths and heights., Electron. Comm. Probab. 9 14-21, 2004
Csaki E., Foldes A., Revesz P.: Maximal local time of a d-dimensional simple random walk on subsets., J. Theoret. Probability 18 : 687-717, 2005
Csaki E., Csorgo M., Rychlik Z., Steinebach J.:: On Vervaat and Vervaat -error type processes for partial sums of renewals., J. Statist. Plann. Inf. J. 137 3-17, 2005
Major P.: An estimate about multiple stochastic integrals with respect to a normalized empirical measure., Studia Scientarum Mathematicarum Hungarica 42: 295-341, 2005
Major P.: An estimate on the supremum of a nice class of stochastic integrals and U-statistics., Probability Theory and Related Fields, 134: 489-537, 2006
Berkes I., Horvath. L.: Convergence of integral functionals of stochastic processes., Econometric Theory 22 304-322, 2006
Berkes I., Weber M.: Almost sure versions of the Darling-Erdos theorem, Stat Probab. Letters 76 280-290, 2006
Berkes I., Aue A, Horvath L.: Strong Approximation for the sums of squares of augmented GARCH sequences, Bernoulli 12 583-608, 2006
Berkes I., Weber M.: On the law of the iterated logarithm for additive functions., Proc. Amer. Math. Soc., 2007
Berkes I., Weber M.: Moment convergence and the law of the iterated logarithm for additive functions, Acta Arithmetica 123 43-55, 2006
Csaki E.: Istvan Vincze (1912-1999) and his contribution to lattice path combinatorics and statistics., J. Statistical Planning and Inference 135: 3-17, 2005
Csaki E.: Mathematical Statistics. A Panorama of Hungarian Mathematics in the Twentieth Century., Bolyai Society Mathematical Studies 14 Springer-Verlag 491-521, 2005
Csaki E., Foldes A. Revesz P., Rosen J., Shi, Z.: Frequently visited sets for random walks., Stochastic Processes and their Applications 115: 1503-1517, 2005
Csaki E., Hu, Y.: On the increments of the principal value of Brownian local time., Electron. Comm. Probab. 10: 925-947, 2005
Major P.: Tail bevahiour of multiple random integrals and U-statistics., Probability Surveys, Vol. 2 448-505, 2005
Berkes I. ,Horvath L.,: Empirical processes of residuals., Empirical process tecchniques for Dependent Data in Birkhauser, Basel 195-209, 2002
Csaki E., Hu, Y.: Length and heights of random walk excursions., Discrete Mathematics and Theoretical Computer Sciencs Discrete Random Walks, DRW03 Conference Volume AC 45-52, 2003
Berkes I. ,Horvath L., Kokoszka, P.: GARCH processes: structure and estimation., Bernoulli, 9 201-227, 2003
Major P.: A multivaraiate generalization of Hoeffding's inequality, Electronic Communication in Probability 2, 220-229, 2006
Revesz. P.: The maximum of the local time of a transient random walk., Studia Sci. Math. Hung. 395--405, 2002
Khosnevisan, D., Revesz P., Shi Z.:: On the explosion of the local times of Brownian sheet alog lines., Ann. Inst. Henry Poincare PR 40 1-24, 2004
Revesz. P.: The maximum of the local time of a transient random walk., Studia Sci. Math. Hung. 395--405, 2004
Revesz. P.: A prediction problem of the branching random walk, J. Applied Probability 41A 25-31, 2004
Revesz. P.: Tell me the values of a Wiener process at integers, I tell its local time., Fields Institute Communications 89-95, 2004
Khosnevisan, D.K., Revesz P.. Shi, Z.:: Level crossings of a two parameter random walk., Stochastic Processes and its applications 112 259-308, 2005
Revesz P., Rosen, J. Shi, Z.:: Large time asymptotics for the density of a branching Wiener process., J. Applied Probability 42 1091-1094, 2005
Csaki, E. , Hu. Y:: Strong Approximations of three-dimensional Wiener sausages., Acta Math. Hungar. 114 205-226, 2007
Csaki, E. , Csorgo M., Rychlik, Z., Steinebach, J:: On Vervaat and Vervaat-error type processes for partial sums and renewals., J. Statist. Plann. Inf. 137 953-966, 2007
Csaki E, Foldes A., Revesz P.:: Heavy points of a d-dimensional simple random walk., Statistics and Probability Letters 76 45-57, 2006
Berkes, I., Horvath. L., Kokoszka. P.: Probabilistic and statistical properties of GARCH processes., Asymptotic Results in Stochastics Frields Instituee Commutiations 44 409-429, 2004
Berkes, I., Horvath. L.,: The efficiency of estimators of paramaters in GARCH processes., Ann. Statist. 43 633-655, 2004
Berkes I. , Csaki E. Csorgo S., Megyesi Z.:: Almost sure limit theorems for sums and maxima from the domain of geometric partial attraction of semistable laws., Limit Theorems in Probability and Statistics, Bolyai Math. Society, 133-157, 2002
Berkes I. ,Horvath L., Kokoszka, P.: Asymptotics for GARCH squared residual correlations, Econometric Theory 19 515-540, 2003
Berkes I. ,Horvath L.,: Approximations for the maximum of stochastic processes with drift., Kibernetika 39 299-306, 2003
Berkes I. ,Horvath L., Kokoszka, P.: Estimation of the moment index of GARCH (1,1) sequence., Econometric Theory 19 565-586, 2003
Berkes I. ,Horvath L.,: Limit results for the empirical process of squared residuals in GARCH models., Stoch. Procc. Appl. 105 271-298, 2003
Berkes I. ,Horvath L.,: Asymptotic results for long memory LARCH sequences., Ann. Appl. Probab.. 13 641-668, 2003
Csaki E., Csorgo M., Foldes A., Shi, Z. Zitikis, R.: Pointwise and uniform asymptotics of the Vervaat error process., J. Theoretical Probability 15 845-875, 2002
Csaki E., Csorgo M., Foldes A., Shi, Z.: On a class of additive functionals of two-dimensional Brownian motion and random walk., Limit Theorems in Probability and Statistics, J. Bolyai Mathematical Society, 321-345, 2002
Csaki E., Hu, E.: On the joint asymptotic behaviours of ranked heighths of Brownian excursions., Limit Theorems in Probability and Statistics, J. Bolyai Mathematical Society, 347-363, 2002
Csaki E., Shi, Z., Yor, M.: Fractional Brownian motions as `higher order' fractional derivatives of Brownian local times., Limit Theorems in Probability and Statistics, J. Bolyai Mathematical Society, 365-387, 2002
Csaki E., Foldes A., Shi, Z.: A joint functional law for the Wiener process and principal value., Studia Sci. Math. Hungar. 40 213-241, 2003
Csaki E., Foldes, A. Hu Y.,: Strong approximations of additive functionals of a planar Brownian motion., Stochastic Processes and their Applications 109, 263-293, 2004



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