Study of the dynamics and structure of sophisticated stochastic systems  Page description

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Details of project

  
Identifier
61052
Type K
Principal investigator Major, Péter
Title in Hungarian Bonyolult sztochasztikus rendszerek dinamikájának és strukturájának vizsgálata
Title in English Study of the dynamics and structure of sophisticated stochastic systems
Keywords in Hungarian Wiener folyamatok funkcionáljai, bolyongások, Garch modellek, sztochasztikus folyamatok nem-lineáris funkcionáljai és azok statisztikai alkalmazásai, véletlen határeloszlástételek, bio-informatika
Keywords in English functionals of Wiener processes, Garch models, non-linear functionals of stochastic processes and their application, random limit theorems, bio-informatics
Discipline
Mathematics (Council of Physical Sciences)100 %
Panel Mathematics and Computing Science
Department or equivalent Alfréd Rényi Institute of Mathematics
Participants Berkes, István
Csáki, Endre
Csiszár, Villő
Tusnády, Gábor
Starting date 2006-02-01
Closing date 2010-01-31
Funding (in million HUF) 15.000
FTE (full time equivalent) 4.16
state closed project
Summary in Hungarian
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\input amstex
\define\summ{\sum\limits}
\nopagenumbers
\noindent
{\it R\"ovid munkaterv}
\medskip\noindent
A Wiener folyamat nem line\'aris funkcion\'aljainak
vizsg\'alata: Nem tipikus ha\-t\'ar\-elosz\-l\'as\-t\'e\-te\-lek,
lok\'alis id\H{o}, tartozk\'od\'asi \'es lok\'alis id\H{o} egy\"uttes
viselked\'es\'enek, fel\'uj\'{\i}t\'asi folyamat
ta\-nul\-m\'a\-nyo\-z\'a\-sa.

\medskip\noindent
F\"uggetlen val\'osz\'{\i}n\H{u}s\'egi v\'altoz\'ok nem
line\'aris funkcion\'aljai, ilyenekre vonat\-ko\-z\'o nagy
elt\'er\'es eredm\'enyek \'es ezek alkalmaz\'asai nem param\'eteres
maximum likelihood becs\-l\'e\-sek\-ben.

\medskip\noindent
ARCH, GARCH \'es anal\'og nemline\'aris id\H{o}sorok
vizsg\'alata. K\"ozgazdas\'agtanban fontos modellek
tanulm\'anyoz\'asa, ezek er\H{o}s \'es gyenge f\"ugg\'ese,
hat\'areloszl\'ast\'etelek, A modellek fontos
\'altal\'anos\'{\i}t\'asai, ezen \'altal\'anos\'{\i}tott
modellek tulajdons\'agai, matematikai
statisztikai alkalmaz\'asokhoz sz\"uks\'eges eredm\'enyek.

\medskip\noindent
A majdnem biztos hat\'areloszl\'ast\'etel. \'Uj tipus\'u
majdnem biztos hat\'areloszl\'ast\'etelek, P\'eterv\'ari
paradoxon, ergod-elm\'eleti m\'odszerek alkalmaz\'asa.
\medskip\noindent
Bio-informatika. Bio-informatikai modellek kidolgoz\'asa,
vizsg\'alata, v\'eletlen gr\'afok Ba\-ra\-b\'a\-si modelle,
\'es annak \'altal\'anos\'{\i}t\'asa, tanul\'o szemin\'arium.



\end
Summary
pc3224 /usr1/users/major-> more short.txt
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{\it Short plan of work}
\medskip\noindent
Study of non-linear functionals of Wiener processes: The study of
non typical limit theorems, local times, joint distribution of local and
occupation type, renewal theory.

\medskip\noindent
Nonlinear functionals of independent random variables large deviation
type estimates for them, and their application in non-parametric
maximum likelihood estimates.

\medskip\noindent
The study of ARCH, GARCH models and similar non-linear time series:
The study of some models important in economy. The investigation of
their weak or strong dependence, limit theorems, important applications
of the models in economical applications, their properties. Results
needed in statistical applications.

\medskip\noindent
Almost sure limit theorems: New type almost sure limit theorems,
Petersburg paradox. Application of ergod-theoretical problems.

\medskip\noindent
Bio informatics: Working out bio-informatic models, the Barab\'asi
model of random graphs and its application, organization of learning
seminars on bio-informatics.



\end




 

Final report

  
Results in Hungarian
Vizsgaltunk Wiener folyamatokat es fuggetlen valoszinusegi valtozok osszeget. Eredmenyeket ertunk el a 3 es magasabb dimenzios Wiener folyamat lokalis idejenek a tanulmanyozasaban es altalanositottuk Erdos es Tayor, Polya, Erdos es Darling eredmenyeit. Bebizonyitottunk nehany a penzugyi matematika altal motivalt statisztikai eredmenyt. E problemak vizsgalata nehany Hilbert ter erteku valoszinusegi valtozokrolszolo nem trivialis feladat megoldasat is kovetelte. Nehany szamelmelettel is kapcsolatos sztochasztikus problemat is vizsgaltunk, ahol az eredmeny bizonyos veletlen jelensegeken kivul egy szamsorozat diophantikus tulajdonsagain is mulott. Ily modon nehany Kac egy klasszikus problemajahoz kapcsolodo uj eredmenyt is kaptunk. Bebizonyitottunk nehany eles becslest U-statisztikakrol es fuggetlen valoszinusegi valtozok mas nem linearis funkcionaljairol. Az ilyen eredmenyeknek erdekes statisztikai alkalmazasai is vannak. Foglalkoztunk a a veletlen permutaciok elmeletevel. Bebizonyitottuk P. McCullagh egy sejteset es erdekes algebrai statisztikai eredmenyeket kaptunk Diaconis egy klasszikus peldajat vizsgalva. Tovabba biostatisztikai motivacioju vizsgalatokat is folytattunk. Tovabbfejlesztettuk Kauffman Boole network-okrol szolo modelljet es nehany immunologiai es nepesseg statisztikai vizsgalatot folytattunk.
Results in English
We investigated Wiener processes and sums of independent random variables. We studied the local time of 3 and higher dimensional Wiener process, and generalized some classical results of Erdos and Tayor, Polya, Erdos and Darling etc. We also studied some new type of statistical problems partly motivated by financial mathematics. These problems demanded the study of some non-trivial problems about random variables taking their values in a Hilbert space. We also studied some problems related to number theory, where the behaviour of some sequences depended both on some random phenomena and diophantic behaviour of some sequences. We got in such a way some new results about a classical problem of Mark Kac. We also got some new sharp estimates about U-statistics and other non-linear functionals of independent random variables. Such results may have interesting statistical applications. We also studied the theory of random permutations. We have proved a conjecture of P. McCullagh and obtained some interesting result on algebraic statistic motivated by a classical example of Diaconis. We also had some other investigations motivated by biostatistics. We developped the model of Kauffman about Boole networks and studied some problems of immunology and population genetics.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=61052
Decision
Yes





 

List of publications

  
Csaki Endre, Csorgo, M. and Rafal, K.: On Vervaat processes for sums of renewals in weakly dependent cases, Dependence in Probability , Analysis and Number Theory. Walter Philipp Memorial Volume. Kendrick Press, 2009
Csaki Endre, Csorgo M., Foldes A., Revesz P.: Random walk local time approxiamated by a Brownian sheet combined with an independent Brownian motion, Ann. Inst. H. Poincare Probab. Statist., 515-544, 2009
Berkes Istvan, Weber, M.: Moment convergence and the law of the iterated logarithm for additive functions., Acta Arithmetica 123, 43--55, 2006
Berkes Istvan, Weber, M.: Almost sure versions of the Darling-Erdos theorem., Probab. Letters 76, 280--290, 2006
Berkes Istvan, Aue, A., Horvath Lajos: Strong approximation for the sums of sqaures of augmented GARCH sequences, Bernoulli, 12, 583--608, 2006
Berkes Istvan, Philipp, W,, Tichy, R.: Empirical Processes in probabilistic number theory: the LIL for the discrepancy of n_k(\omega) mod 1, Illinois J. Math. 50, 107-145, 2006
Berkes Istvan, Horvath Lajos: Convergence of integral functionals of stochastic processes., Econometric Theory 22, 304--322, 2006
Major Peter: A multivariate generalization of Hoeffding's inequality., Electronic Communication in Probability 2, 220--229, 2006
Csaki Endre, Foldes Antonia, Revesz Pal: Joint asymptotic behavior of local and occupation times of random walk in higher dimension., Studia Sci. Math. Hungar. 44 535--563, 2007
Berkes Istvan, Horvath Lajos, Kokoszka, P., Shao, Q., M.:: On discriminating betweeen longrange dependence and changes in the mean., Annals of Statistics 34, 1140--1165, 2006
Csaki Endre, Foldes Antonia, Revesz Pal: On the local times of transient random walks., Acta Applicandae Mathematicae 96 147--158, 2007
Tusnady Gabor, Rejto Lidia: Reconstruction of Kauffman networks applying trees., Linear Algebra and its applications 417, 220--244, 2006
Csaki Endre, Hu, Y.: Strong approximations of three-dimensional Wiener sausages., Acta Sci. Math. Hungar. 114 205-226, 2007
Berkes Istvan, Weber, M.: On the convergence of $\sum c_kf(n_kx)$., Memoirs of the AMS 201, no 943, 1-72, 2009
Berkes Istvan, Weber, M.: On the law of the iterated logarithm for arithmetic functions, Proc. Amer. Math. Soc. 135 1223 1232, 2007
Berkes Istvan, Weber, M.: On the convergence of $\sum c_kf(n_kx)$., Memoirs of the AMS, 2006
Berkes Istvan, Philipp W., Tichy R.: Pseudorandom numbers and entropy conditions, J. of Complexity 23 516-527, 2007
Berkes Istvan, Plilipp W., Tichy, R.: Metric discrapency results for sequences $\{n_kx\}$ and diophantine equations, Diophantine Approximations (eds. H. P. Schlickewei, K. Schmidt and R. Tichy) 95-105 (Springer), 2008
Berkes Istvan, Weber, M.: On the complete convergence of triangular arrays of independent random variables, Stat. Probab.Letters 77 952-963, 2007
Csaki Endre, Foldes, A. Revesz P.: On the behaviour of rendom walk around heavy points, J. Theoret. Probab. 20 1041--1057, 2007
Major P.: On a multivariate version of Bernstein's inequality., Electronic Journal of Probability 12 966--988, 2007
Tusnady G., Vizkelety, Gy., Petho, B,, Czenner, Zs,, Vargha, A,, Meggyes, K.: Intellectual deficit in endogenous psyshoses: 21--33 year follow up a prospective study ("Budapest 2000"), Neurology, Pschiatry and Brain research 87 19--26, 2007
Tusnady G., Petho, B,, Vargha, A,, Tolna, M. Farkas, Gy. Vizkelety, A,, Toth, A,Szilagyi , I., Bitter, I., Kelemen, A., Czobor, P.,: Validity and reliability: comparison of inter-rater reliabilities of psychopathologiccal symptoms., The Journal of Nervous and Mental Disease 195/7 606-613, 2007
Tusnady G., Bakacs, T., Mehrishi, J. N., Szabados, T., Varga, L., Szabo, M.,: T cells survey the stability of the self: a testable hypothesis on the homeostatic role of TCR--MHC interactions., International Archives of Allergy and Immunology, 144 171--182, 2007
Csiszar, V., Rejto, L., Tusnady, G.,: Statistical inference on random structures, Horizon of Combinatorics, Balatonalmadi 2006, Bolyai Society Mathematical Studies, X. 1--30, 2007
Berkes Istvan, Aue. A., Horvath Lajos: A note on the existence of solutions of stochastic recurrene equations, Acta Sci. Szeged 73 767-779, 2007
Berkes Istvan, Aue. A., Horvath Lajos: Selection from a stable box, Bernoulli 14 125-139, 2008
Berkes Istvan, Plilipp W., Tichy, R.: Entropy conditions for subsequences of random variables with application to empirical processes, Monatshefte Math. 153 (183-204), 2008
Berkes Istvan, Hormann, S. Horvath Lajos: The functional central limit theorem for a class of GARCH observations with applications, Stat. Probab. Letters 78 2725-2730, 2008
Berkes Istvan, Aistleitner, C.: On the law of the iterated logarithm for the discrepancy of $n_kx$, Monatshefte Math. 156 103-121, 2009
Berkes Istvan, Gombai Edit, Horvath Lajos: Testing for the changes in the covariance structure of linear processes, J. Stat. Planning and Inf. 139 2044-2063, 2009
Berkes Istvan, Gabrys R., Horvath Lajos, Kokoszka, P.: Detecting for changes in the mean of functional observations, J. Royal Stat. Soc. B 71,927-941, 2009
Berkes Istvan, Hormann, S, Schauer, J.: Asymptotic results for the empirical processes of stationary sequences, Stoch. Proc. Appl. 119, 1298-1324, 2009
Berkes Istvan, Horvath Lajos, Hormann, S,: On the functional versions of the arc sine laws, J. Theoretical Probability, 2008
Berkes Istvan, Aistleitner, C.: On the central limit theorem for $f(n_kx)$, Probability Theory Related Fields 146, 267-289, 2009
Berkes Istvan, Horvath Lajos, Schauer, J.: Bootstrap and permutation statistics under infinite variance, Statistics for Dependent Data (eds. Paul Doukhan, Donatas Surgailis, Gilles Teyssire) Lecture Notes in Statistics, 2009
Csaki Endre, Foldes, A. Revesz P.: On the local time of the asymmetric random walk, Acta Sci. Math. Szeged 74 349-379, 2008
Csaki Endre, Foldes, A. Revesz P.: Transient nearest neighbor walk on the line, J. Theoretic Probab. 22, 992-1009, 2009
Csaki Endre, Foldes, A. Revesz P.: Transient nearest neighbor walk and Bessel process, J. Theoretic Probab., 2009
Csaki Endre, Foldes, A. Revesz P.: Strong limit theorems for a simple random walk on the 2-dimensional comb., Electr. J. Probab. 14, 2391-2417, 2009
Berkes Istvan, Horvath Lajos, Schauer, J.: Noncentral limit theorem for random selections, J. Prob. Theory Rel. Fields, 2008
Berkes Istvan, Hormann, S., Weber, M.: Upper-lower tests for weighted i.i.d. sequences and martingales, J. Theor. Prob., 2009
Berkes Istvan, Horvath Lajos, Lin, S.: Estimation in random coefficient autoregressive models, Journal of Time Series Analysis. 30, 359-416, 2009
Csiszar V.: Conditional independence relations and log-linear models for random matchings, Acta Math. Hungar. 122, 131-152, 2009
Csiszar V.: Markov bases for conditional independence models for permutations, Kybernetika 45 249-260, 2009
Csiszar V.: Jump Home and Shift: An Acyclic Operation on Permutations, Amer. Math. Monthly, 2009
Tusnady Gabor, Petho B. Tolna, J., Farkas M., Vizkelety Gy., Varga A., Czobor P.: The predictive validity of the Leonhardean classification of endogenous psychoses: a 21-33 follow-up of a prospective study, European Archives of Psycioatry and Clinical Neuroscience 258 324-334, 2008
Tusnady Gabor, Smith, J. L., Barrett, L., Rejto L., Carry, S. C.: Resolving environmental drivers of microbial diversity in antarctic soils, Polar Science (37) 56-76, 2008
Tusnady Gabor, Hatvani L. Tokos G.: A mutation-selection recombination model in population genetics, Dynamical Systems and its application 42 79-98, 2008
Tusnady Gabor, Gaudi I., Rejto L., Kasler M., Szentirmay Z.: A magyar daganatos betegek tulelesi eselyei a nemzeti rakregiszter adatai alapjan, Magyar Onkologia 52 339-349, 2008



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