Some problems of modern probability theory  Page description

Help  Print 
Back »
 

Details of project

  
Identifier
81928
Type K
Principal investigator Major, Péter
Title in Hungarian A modern valoszinusegszamitas nehany problemaja
Title in English Some problems of modern probability theory
Keywords in Hungarian ARMA modszerek, valoszinusegszamitas Hilbert terekben, valoszinusegszamitasi modszerek az analizisben es szamelmeletben, Wiener-Ito integralok, empirikus folyamatok nem linearis funkcionaljai,
Keywords in English ARMA methods, probability in Hilbert spaces, probabilistic methods in the analysis and number theory, Wiener-iIto integrals, non-linear functionals of empirical processes
Discipline
Mathematics (Council of Physical Sciences)100 %
Ortelius classification: Probability theory
Panel Mathematics and Computing Science
Department or equivalent Alfréd Rényi Institute of Mathematics
Participants Berkes, István
Pham Ngoc, Ánh
Starting date 2010-02-01
Closing date 2013-12-31
Funding (in million HUF) 4.384
FTE (full time equivalent) 3.88
state closed project
Summary in Hungarian
Gauss polinomok es U-statisztikak farokeloszlasara kivanunk jo
becsleseket adni. Megjelentek rangos folyoiratokban olyan cikkek, amelyek
hibas eredmenyeket tartalmaznak. Ezeket ki kivanjuk javitani, es a
problemakorrol jol ertheto, teljes kepet akarunk adni. E kerdesekkel azert
foglalkozunk behatobban, mert e vizsgalatok eredmenyei lehetove teszik uj,
hatekony modszerek kidolgozasat nemparameteres maximum likelihood becslesek
es fuggetlen valoszinusegi valtozok nem linearis funkcionaljanak vizsgalataban.

Nehany olyan statisztikai problemat vizsgalunk, amelyek olyan nepszeru
modszerek alkalmazasaban jelentek meg varatlanul, mint a bootstrap eljaras.
Ez jo becslest ad, ha a minta veges szorasu elemekbol all. De ez a feltetel
tobb fontos feladatban nem teljesul. Ilyenkor a nagy erteku mintaelemek
elhagyasaval probaljuk a nehezseget athidalni. De kiderult, hogy a helyzet
sokkal komplexebb, mint azt eredetileg gondoltak. Az itt felmerulo kerdesek
komolyabb vizsgalatot igenyelnek.

Egy masik kutatasi tema minden koordinatajaban sok adatot tartalmazo
rendszerek vizsgalataban jelent meg. Ilyenkor erdemes Hilbert ter erteku ARMA
modelleket vizsgalni, es az ertekeit egy vegtelen dimenzios terben felvevo
valoszinusegi valtozokat egy olyan kisebb terbe vetiteni, amely megorzi a
lenyeges informaciokat. E vetites soran a fuggetlensegi tulajdonsagok
gyengulnek, es a modell sikeres vizsgalata erdekeben meg kell talalni a
rendszer jo keveresi tulajdonsagait.
Egy harmadik tema olyan valoszinusegi problemak vizsgalata, amelyeknek az
analizisben van jelentoseguk. Ismeretes, hogy egy $L_p$ terben korlatos
fuggvenysorozatnak van olyan reszsorozata, amely sok tekintetben ugy
viselkedik, mint fuggetlen valoszinusegi valtozok sorozata. Finomabb
vizsgalatok kideritettek, hogy az itt felmerulo jelensegek olyan szamelmeleti
problemakhoz kapcsolodnak, mint a diofantikus egyenletek viselkedese.
Hincsin egy regi klasszikus problemajanak az altalanositasaval is
foglalkozni kivanunk, mely arrol szol, hogy egy periodusu integralhato
$f(x)$ fuggvenyekre az $f(nx)$ fuggveny sorozat mely reszsorozatainak
az atlaga viselkedik ugy, ahogy az ergod tetel alapjan varjuk.
Summary
We want to give good estimates on the tail behavior of Gaussian polynomials and U-statistics. Some papers appeared in respected journals that contain incorrect results in this field. We want to correct them and to give a good complete picture about this problem. One cause of our study is that these results enable us to work out new strong methods in the study of non-parametric statistical problems and of limit theorems for non-linear functionals of independent random variables.

We study statistical problems which appeared in the study of such popular methods as the bootstrap. It gives a good estimate if the sample consists of random variables with finite variance. But this condition is not satisfied in some important models. In such cases we try to overcome the difficulty by omitting large data. But it turned out that the problem is much harder than we had expected. These problems demand serious investigation.

Another topic is the study of systems containing many data in all coordinates. In such cases it is worth studying Hilbert space valued ARMA models, and to project infinite dimensional random variables to such finite dimensional spaces that preserve the important information. During this projection the independence properties get weaker, and to make a successful study of the models we have to find the good mixing properties of the system.

We also study such probability problems which have importance in analysis. It is known that all $L_p$ bounded sequences of functions have a subsequence which behaves similarly to sums of independent random variables. Finer studies showed that these questions are related to such number theoretic problems as the behavior of diophantine equations. We want to deal with a generalization of a classical problem of Hinchin. It deals with the problem that given a 1 periodic in [0,1] integrable function $f(x)$ the average of which subsequences of $f(nx)$ show such a behavior that the ergod theorem suggests.




 

Final report

  
Results in Hungarian
Kutatasaink 4 temat tartalmaznak. 1.) Olyan problemakat neztunk, amelyekben a rendszernek mind a valoszinusegi mind a szamelmeleti tulajdonsagai fontosak. Azt vizsgaltuk, hogy bizonyos hezagos sorok, pl. hezagos Fourier sorok mikor viselkednek fuggetlen valoszinusegi valtozok osszegeihez hasonloan. A valasz fugg az egyutthatok szamelmeleti tulajdonsgaitol. 2.) Olyan statisztikai problemakat vizsgaltunk, ahol a hagyomanyos modszerek nem alkalmazhatoak. Ilyenek pl. a vegtelen szorasu valoszinusegi valtozok osszegei, amelyekre nem ervenyesek a klasszikus tetelek. Megmutattuk, hogy ilyen esetekben a nagy tagok elhagyasaval, az un. trimming modszerrel jo becslesek adhatok, es bebizonyitottuk az ehhez szukseges eredmenyeket. 3.) Alkalmas stacionarius sorozatok jo kozeliteset adtuk meg Wiener folyamatok segitsegevel olyan esetekben is, amikor a Komlos-Major-Tusnady approximacio nem alkalmazhato. 4.) Eles becsleseket adtunk normalizalt empirikus mertekek szerinti tobb-valtozos integralokra es ilyen integralok szupremumara. Megmutattuk, hogy ezek az eredmenyek alkalmazhatoak nehezen vizsgalhato hatareloszlastetelek vizsgalataban. Errol az elmeletrol Lecture Note-ot irtunk, amelyet a Springer kiado megjelentetett.
Results in English
We made research on 4 subjects. 1.) We proved results where both the probabilistic and number theoretic properties of the system play important role. We investigated when some rarified series, e.g. rarified trigonometrical series behave similarly to sums of independent random variables. The answer depends on the number theoretical properties of the coefficients. 2.) We studied statistical problems where the classical methods do not work because the very large value of some terms. We considered such cases when the random terms have infinite second moments, hence the limit theorems behind classical methods do not hold for them. We presented such methods with the help of a good trimming which work even in such cases, and proved the theoretical results which guarantee this. 3.) We gave a good approximation of certain stationary processes with the help of Wiener processes in cases when the classical Komlos-Major-Tusnady method does not work. 4.) We estimated multiple integrals with respect to empirical measures. We took the normalization of empirical measures and its direct product with itself. We gave good estimate on the tail distribution of an integral with respect to this measure. We also gave sharp estimate on the supremum of such integrals for a class of kernel functions. We showed that these results are useful in the study of hard limit theorems. We wrote a lecture note on this subject, which the Springer Verlag published.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=81928
Decision
Yes





 

List of publications

  
Anh, Pham, Ngoc, Wyk van L.: Isomorhisms between strongly triangular matrix rings, Linear algebra and its applications, 438, 4374-4381, 2013
Major, P.: Multiple Wiener--Ito integrals with applications to limit theorems, Springer Lecture Notes in Mathematics, 849, Second edition, 124 pages, 2014
Berkes I., Horvath L., Schauer, J.,: Permutation and bootstrap statistics under infinite variance, Lecture Notes in Statistics, Vol. 200. pp 1-20, Springer, 2010
Berkes, I., Horvath, L., Schauer, J.,: Asymptotic behavior of trimmed CUSUM statistics, Bernoulli Journal, 17, 1344 -1367, 2011
Berkes, I., Horvath, L., Schauer, J.,: Asymptotic behavior of trimmed sums, Stochastic and Dynamics 12, #1150002, 2012
Aistleitner, C., Berkes, I., Tichy, R.,: On permutations of the Hardy-Littlewood-Polya sequences, Trans. Amer. Math. Soc., 363, 6219-6244, 2011
Aistleitner, C., Berkes I.: Probability and metric discrepancy theory, Stochastics and Dynamics 11, 183 - 207., 2011
Berkes, I., H"ormann, S., Schauer, J.: Split Invariance principles for stationary processes, Annals of Probability 39 pp. 2441-2473, 2011
Aistleitner, C., Berkes, I., Tichy, R.: On the asymptotic behavior of weakly lacunary sequences, Poc. Amer. Math. Soc., 139, 2505 - 2517, 2011
Aistleitner, I., Berkes, I., Tichy, R.: On the law of the iterated logarithm for permuted lacunary sequences., Proceeedings of the Steklov Institute of Mathematics, 276, 3--20, 2012
Berkes, I., Horvath, I.: The central limit theorem for sums of trimmed variables with heavy tails, Stoch. Proc. Appl. 122 449-465, 2012
Aistleitner, C., Berkes, I.: Limit distributions in metric discrepancy theory, Monatshefte Math., 2011
Alstleitner, R, Berles, I., Tichy, R.: On the permutations of lacunary series, RIMS Kokuuroku Bessatsu, B34 1--34, 2012
Berkes, I. ,M\"uller, W. and Weber, M.: On the law of large numbers and arithematic functions, Indigationes Math. 23, 547--555, 2012
Berkes, I. Horvath, G and Rice, G.: Weak invariance principles for sums of dependent random functions, Stoch. Proc. Appl. 123, 385--403, 2013
Berkes, I. and Wu, W., B.: Komlos--Major--Tusnady approximation under dependence, Ann. Probability, to appear, 2014
Berkes, I. and Weber, M.: On series of dilated functions, Quaterly J. Math., to appear, 2014
Berkes, I.: Change point detection with stable AR(!) errors, Asymptotic methods in stochastics, Festschrift for M. Csorgo (to appear), 2014
Major, Peter: On the estimation of multiple random integrals and $U$-statistics, Lecture Notes Series of Springer Verlag 2079, 298 p., 2013
Major, P.,: Estimation of multiple random integrals and U-statistics, Moscow Mathematical Journal, Vol. 10, No 4. October-December 2010, 747-767, 2010




 

Events of the project

  
2011-10-25 12:37:50
Résztvevők változása



Back »