Asymptoticmethods in stochastics  Page description

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Details of project

  
Identifier
43037
Type K
Principal investigator Csáki, Endre
Title in Hungarian Aszimptotikus módszerek a sztochasztikában
Title in English Asymptoticmethods in stochastics
Panel Mathematics and Computing Science
Department or equivalent Alfréd Rényi Institute of Mathematics
Participants Berkes, István
Móri, Tamás
Révész, Pál
Starting date 2003-01-01
Closing date 2007-12-31
Funding (in million HUF) 8.058
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
A véletlen bolyongás témakörében vizsgáltuk tranziens esetben a lokális idő tulajdonságait, különös tekintettel a sokszor meglátogatott pontokra, ill. azok környezetére. Ezen eredmények Erdős és Taylor klasszikus eredményeit élesítik, ill. terjesztik ki. A közgazdasági matematikában fontos szerepet játszó ARCH és GARCH folyamatok és általánositásaik valószinűségszámítási és statisztikai tulajdonságaival foglalkoztunk, többek között a paraméterbecslések, konzisztencia problémák, határeloszlások és nemparaméteres módszerek területén. Pszeudovéletlen számokkal kapcsolatban vizsgáltuk azok egyenletességét egy új mértékszám, az ún. well-distribution measure alapján. Analízisbeli módszerek segitségével meghatároztuk néhány klasszikus pszeudovéletlen konstrukció diszkrepanciáját is. Véletlen fákkal kapcsolatban a Barabási-Albert modellt és annak különböző általánositásait vizsgáltuk, azokra fokszámeloszlást, valamint a maximális fokszám tulajdonságait határoztuk meg. Erős invariancia tételeket adtunk meg a bolyongás kirándulásainak hosszára és magasságára, a kétdimenziós Wiener folyamat additív funkcionáljaira, valamint a háromdimenziós Wiener folyamat trajektóriája körüli tartományra (Wiener sausage).
Results in English
For transient random walks we investigated the properties of local times, in particular the frequently visited points and their neighbors. These extend the classical results of Erdős and Taylor. We investigated the probabilistic and statistical properties of ARCH and GARCH processes, playing an important role in financial mathematics and econometrics. In particular, we proved several results for the parameter estimation, consistency problems, limit distributions and nonparametric procedures for such processes. Concerning pseudorandom numbers, we investigated their uniformity, based on the so-called well-distribution measure. Using methods of analysis, we studied also the discrepancy of pseudorandom constructions. For random trees we investigated the Barabási-Albert model and its generalizations. We determined degree distributions and the properties of maximal degree. We have established strong invariance principles for lengths and heights of random walk excursions, additive functionals of two-dimensional Wiener process and the Wiener sausage in three dimension.
Full text http://real.mtak.hu/889/
Decision
Yes





 

List of publications

  
Berkes I; Horváth L; Kokoszka P.: GARCH processes: structure and estimation, Bernoulli 9: 201-227, 2003
Berkes I; Horváth L.: Asymptotic results for long memory LARCH sequences, Ann. Appl. Probab. 13: 641-668, 2003
Berkes I; Horváth L.: The rate of consistency of the quasi-maximum likelihood estimator, Statist. Probab. Lett. 61: 133-143, 2003
Berkes I; Horváth L; Kokoszka P.: Estimation of the maximal moment exponent of a GARCH(1,1) sequence, Econometric Theory 19: 565-586, 2003
Berkes I; Horváth L.: Approximations for the maximum of stochastic processes with drift, Kybernetika (Prague) 39: 299-306, 2003
Csáki E; Földes A; Shi Z.: A joint functional law for the Wiener process and principal value, Studia Sci. Math. Hungar. 40: 213-241, 2003
Csáki E; Hu Y.: Lengths and heights of random walk excursions, In: Discrete Mathematics and Theoretical Computer Science, Discrete Random Walk Conf., Paris 2003. pp. 45-52, 2003
Révész P.: Tell me the values of a Wiener process at integers, I tell you its local time, In: Asymptotic Methods in Stochastics, Fields Institute Communications, Vol. 44, pp. 89-95, 2004
Berkes I; Horváth L.: The efficiency of the estimators of the parameters in GARCH processes, Ann. Statist. 32: 633-655, 2004
Berkes I; Horváth L; Husková M; Steinebach J.: Applications of permutations to the simulations of critical values, J. Nonparametr. Statist. 16: 197-216, 2004
Berkes I; Horváth L; Kokoszka P.: Probabilistic and statistical properties of GARCH processes, In: Asymptotic Results in Stochastics, Fields Institute Communications, Vol. 44, pp. 409-429, 2004
Csáki E; Révész P; Shi Z.: Large void zones and occupation times for coalescing random walks, Stoch. Process. Appl. 111: 97-118, 2004
Csáki E; Hu Y.: On the ranked excursion heights of a Kiefer process, J. Theoret. Probab. 17: 145-163, 2004
Csáki E; Földes A; Shi Z.: Our joint work with Miklós Csörgő, In: Asymptotic Methods in Stochastics, Fields Institute Communications, Vol. 44, pp. 3-24, 2004
Révész P.: A prediction problem of the branching random walk, J. Appl. Probab. 41A: 25-31, 2004
Berkes I; Weber M.: Upper and lower class tests along subsequences, Stoch. Process. Appl., 115: 679-700, 2005
Berkes I; Horváth L; Kokoszka P.: Near integrated GARCH sequences, Ann. Appl. Probab. 15: 890-913, 2005
Berkes I; Horváth L; Kokoszka P.: Testing for parameter constancy in GARCH(p,q) models, Statist. Probab. Lett. 70: 263-273, 2005
Berkes I; Horváth L; Kokoszka P; Shao QM.: On discriminating between long-range dependence and changes in the mean, Ann. Statist. 34: 1140-1165, 2006
Csáki E; Földes A; Révész P.: Maximal local time of a d-dimensional simple random walk on subsets, J. Theoret. Probab. 18: 687-717, 2005
Csáki E; Csörgő 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
Móri TF.: The maximal degree of the Barabási-Albert random tree, Combin. Probab. Computing 14: 339-348, 2005
Csiszár V; Móri TF; Székely GJ.: Chebyshev-type inequalities for scale mixtures, Statist. Probab. Lett. 71: 323-335, 2005
Khoshnevisan D; Révész P; Shi Z.: Level crossings of a two-parameter random walk, Stoch. Process. Appl. 115: 359-380, 2005
Csáki E.: István Vincze (1912-1999) and his contribution to lattice path combinatorics and statistics, J. Statistical Planning and Inference 135: 3-17, 2005
Csáki E; Hu Y.: On the increments of the principal value of Brownian local time, Electronic J. Probability 10: 925-947, 2005
Csáki E; Földes A; Révész P; Rosen J; Shi Z.: Frequently visited sets for random walks, Stochastic Processes and their Applications 115: 1503-1517, 2005
Csáki E; Földes A; Révész P.: Heavy points of a d-dimensional simple random walk, Statistics and Probability Letters 76: 45-57, 2006
Móri TF.: A surprising property of the Barabási-Albert random tree, Studia Sci. Math. Hungar. 43: 263-271, 2006
Móri TF.: Another class of scale free random graphs, 25th European Meeting of Statisticians (24-28 July 2005, Oslo), Final Programme and Abstracts, p. 256, 2005
Révész P; Rosen J; Shi Z.: Large time asymptotics for the density of a branching Wiener process, J. Applied Probability 42: 1081-1094, 2005
Berkes I; Horváth L; Kokoszka P; Shao QM.: Almost sure convergence of the Bartlett estimator, Periodica Math. Hungar. 51: 11-25, 2005
Berkes I; Weber M.: Almost sure versions of the Darling-Erdős theorem, Statist. Probab. Lett., 76: 280-290, 2006
Berkes I; Horváth L.: Convergence of integral functionals of stochastic processes, Econometric Theory, 22: 304-322, 2006
Aue A; Berkes I; Horváth L.: Strong approximation for the sums of squares of augmented GARCH sequences, Bernoulli 12: 583-608, 2006
Berkes I; Weber M.: Moment convergence and the law of the iterated logarithm for additive functions, Acta Arithmetica 123: 43-55, 2006
Berkes I; Philipp W; Tichy R.: Empirical processes in probabilistic number theory: the LIL for the discrepancy, Illinois J. Math. 50: 107-145, 2006
Katona Zs; Móri TF.: A new class of scale free random graphs, Statist. Probab. Lett. 76: 1587-1593, 2006
Berkes I; Philipp W; Tichy R.: Pseudorandom numbers and entropy conditions, J. Complexity 23: 516-527, 2007
Berkes I; Philipp W; Tichy R.: Metric discrepancy results for sequences $\{n_k x\}$ and diophantine equations, Diophantine Approximation. Festschrift for W. M. Schmidt, Ed.: R. Tichy, Springer (közlésre elfogadva), 2008
Berkes I; Weber M.: On the convergence of $\sum c_k f(n_kx)$, Memoirs of the AMS (közlésre elfogadva), 2008
Berkes I; Weber M.: On complete convergence of triangular arrays of independent random variables, Statist. Probab. Lett. 77: 952-963, 2007
Csáki E; Hu Y.: Strong approximations of three-dimensional Wiener sausages, Acta Sci. Math. Hungar. 114: 205-226, 2007
Csáki E; Földes A; Révész P.: On the local times of transient random walks, Acta Applicandae Mathematicae 96: 147-158, 2007
Csáki E; Földes A; Révész P.: Joint asymptotic behavior of local and occupation times of random walk in higher dimension, Studia Sci. Math. Hungar. 44: 535-563, 2007
Csáki E; Földes A; Révész P.: On the behavior of random walk around heavy points, J. Theoret. Probab. 20: 1041-1057, 2007
Aue A; Berkes I; Horváth L.: Selection from a stable box, Bernoulli (közlésre elfogadva), 2008
Aue A; Berkes I; Horváth L.: A note on the existence of solutions of stochastic recurrence equations, Acta Sci. Math. (Szeged) (közlésre elfogadva), 2008
Berkes I; Philipp W; Tichy R.: Entropy conditions for subsequences of random variables with applications to empirical processes, Monatshefte Math. (közlésre elfogadva), 2008
Aistleitner C; Berkes I.: On the law of the iterated logarithm for the discrepancy of ${n_kx}$, Monatshefte Math. (közlésre elfogadva), 2008
Csáki E; Földes A; Révész P.: On the local time of the aszmmetric Bernoulli walk, Acta Sci. Math. (Szeged) (közlésre elfogadva), 2008
Berkes I; Horváth L.: Limit results for the empirical process of squared residuals in GARCH models, Stoch. Process. Appl. 105: 271-298, 2003
Berkes I; Horváth L; Kokoszka L.: Asymptotics for GARCH squared residual correlations, Econometric Theory 19: 515-540, 2003
Berkes I; Gombay E; Horváth L; Kokoszka P.: Sequential change-point detection in GARCH(p,q) models, Econometric Theory 20: 1140-1167, 2004
Csáki E; Földes A; Hu Y.: Strong approximations of additive functionals of a planar Brownian motion, Stoch. Process. Appl. 109: 263-293, 2004
Csáki E; Hu Y.: Invariance principles for ranked excursion lengths and heights, Electron. Comm. Probab. 9: 14-21, 2004
Csiszár V; Móri TF.: The convexity method of proving moment-type inequalities, Statist. Probab. Lett. 66: 303-313, 2004
Khoshnevisan D; Révész P; Shi Z.: On the explosion of the local times along lines of Brownian sheet, Ann. Inst. H. Poincaré 40: 1-24, 2004
Révész P.: The maximum of the local time of a transient random walk, Studia Sci. Math. Hungar. 40: 395-406, 2004
Berkes I; Weber M.: On the law of the iterated logarithm for additive functions, Proc. Amer. Math. Soc. 135: 1223-1232, 2007
Móri TF.: On a 2-parameter class of scale free random graphs, Acta Sci. Math. Hungar. 114: 37-48, 2007



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