TY - JOUR AU - Berkes, István AU - Siegfried, Hörmann TI - Some optimal conditions for the ASCLT JF - JOURNAL OF THEORETICAL PROBABILITY J2 - J THEOR PROBAB VL - 37 PY - 2024 SP - 209 EP - 227 PG - 19 SN - 0894-9840 DO - 10.1007/s10959-023-01245-w UR - https://m2.mtmt.hu/api/publication/33670168 ID - 33670168 N1 - Published online: 06 May 2023 LA - English DB - MTMT ER - TY - JOUR AU - Christoph, Aistleitner AU - Berkes, István AU - Robert, Tichy TI - Lacunary sequences in analysis, probability and number theory JF - LECTURE NOTES IN MATHEMATICS J2 - LECT NOTES MATH VL - accepted PY - 2024 SP - & SN - 0075-8434 UR - https://m2.mtmt.hu/api/publication/33670095 ID - 33670095 LA - English DB - MTMT ER - TY - JOUR AU - Berkes, István AU - Borda, Bence TI - Random walks on the circle and Diophantine approximation JF - JOURNAL OF THE LONDON MATHEMATICAL SOCIETY J2 - J LOND MATH SOC VL - 108 PY - 2023 IS - 2 SP - 409 EP - 440 PG - 32 SN - 0024-6107 DO - 10.1112/jlms.12749 UR - https://m2.mtmt.hu/api/publication/34236327 ID - 34236327 N1 - Export Date: 28 February 2024 Correspondence Address: Borda, B.; Graz University of Technology, Steyrergasse 30, Austria; email: borda@math.tugraz.at Funding details: Austrian Science Fund, FWF, Y 901 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFIH, K 125569 Funding text 1: István Berkes is supported by NKFIH grant K 125569. Bence Borda is supported by the Austrian Science Fund (FWF) project Y 901. AB - Random walks on the circle group R/Z whose elementary steps are lattice variables with span alpha is not an element of Q or p/q is an element of Q taken mod Z exhibit delicate behavior. In the rational case, we have a random walk on the finite cyclic subgroup Z(q), and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper, we extend these results to random walks with irrational span alpha, and explicitly describe the transition of these Markov chains from finite to general state space as p/q -> alpha along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a phase transition from polynomial to exponential decay after approximate to q(2) steps. This seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purely exponential. LA - English DB - MTMT ER - TY - JOUR AU - Bazarova, Alina AU - Berkes, István AU - Horváth, Lajos TI - Trimmed Least Square Estimators for Stable Ar(1) Processes JF - MATHEMATICA PANNONICA J2 - MATH PANNONICA VL - 28_NS2 PY - 2022 IS - 1 SP - 16 EP - 23 PG - 8 SN - 0865-2090 DO - 10.1556/314.2022.00003 UR - https://m2.mtmt.hu/api/publication/34845921 ID - 34845921 AB - We prove the weak consistency of the trimmed least square estimator of the covariance parameter of an AR(1) process with stable errors. LA - English DB - MTMT ER - TY - JOUR AU - Berkes, István AU - Csáki, Endre TI - On the Almost Sure Central Limit Theorem Along Subsequences JF - MATHEMATICA PANNONICA J2 - MATH PANNONICA VL - 28_NS2 PY - 2022 IS - 1 SP - 11 EP - 15 PG - 5 SN - 0865-2090 DO - 10.1556/314.2022.00002 UR - https://m2.mtmt.hu/api/publication/34845912 ID - 34845912 AB - Generalizing results of Schatte [11] and Atlagh and Weber [2], in this paper we give conditions for a sequence of random variables to satisfy the almost sure central limit theorem along a given sequence of integers. LA - English DB - MTMT ER - TY - JOUR AU - Berkes, István AU - Borda, Bence TI - On the discrepancy of random subsequences of [n alpha}, II JF - ACTA ARITHMETICA J2 - ACTA ARITH VL - 199 PY - 2021 IS - 3 SP - 303 EP - 330 PG - 28 SN - 0065-1036 DO - 10.4064/aa200811-25-1 UR - https://m2.mtmt.hu/api/publication/32379817 ID - 32379817 N1 - Funding Agency and Grant Number: NKFIH grantNational Research, Development & Innovation Office (NRDIO) - Hungary [K 125569]; Austrian Science Fund (FWF)Austrian Science Fund (FWF) [Y-901] Funding text: Research of I. B. was supported by NKFIH grant K 125569.; Research of B. B. was supported by the Austrian Science Fund (FWF), project Y-901. LA - English DB - MTMT ER - TY - JOUR AU - Berkes, István AU - Borda, Bence TI - On the discrepancy of random subsequences of {n alpha} JF - ACTA ARITHMETICA J2 - ACTA ARITH VL - 191 PY - 2019 IS - 4 SP - 383 EP - 415 PG - 33 SN - 0065-1036 DO - 10.4064/aa180417-12-12 UR - https://m2.mtmt.hu/api/publication/30945205 ID - 30945205 N1 - Funding Agency and Grant Number: NKFIH grant [K 125569] Funding text: Research of I. Berkes was supported by NKFIH grant K 125569. Export Date: 8 February 2020 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, NKFI, K 125569 Funding text 1: Research of I. Berkes was supported by NKFIH grant K 125569. AB - Abstract For irrational α, {nα} is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences {nkα}, with the exception of metric results for exponentially growing (nk). It is therefore natural to consider random (nk), and in this paper we give nearly optimal bounds for the discrepancy of {nkα} in the case when the gaps nk+1−nk are independent, identically distributed, integer valued random variables. As we will see, the discrepancy behavior is determined by a delicate interplay between the distribution of the gaps nk+1 − nk and the rational approximation properties of α. We also point out an interesting critical phenomenon, i.e. a sudden change of the order of magnitude of the discrepancy of {nkα} as the Diophantine type of α passes through a certain critical value. LA - English DB - MTMT ER - TY - JOUR AU - Alina, Bazarova AU - Berkes, István AU - Marko, Raseta TI - On the discrepancy of random walks on the circle JF - UNIFORM DISTRIBUTION THEORY J2 - UNIF DISTRIB THEOR VL - 14 PY - 2019 IS - 2 SP - 73 EP - 86 PG - 14 SN - 1336-913X DO - 10.2478/udt-2019-0015 UR - https://m2.mtmt.hu/api/publication/30803174 ID - 30803174 LA - English DB - MTMT ER - TY - JOUR AU - Berkes, István AU - Borda, Bence TI - On the law of the iterated logarithm for random exponential sums JF - TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY J2 - T AM MATH SOC VL - 371 PY - 2019 IS - 5 SP - 3259 EP - 3280 PG - 22 SN - 0002-9947 DO - 10.1090/tran/7415 UR - https://m2.mtmt.hu/api/publication/3414600 ID - 3414600 N1 - Accepted in November 2017, Published electronically: December 7, 2018 Export Date: 4 October 2019 Funding details: Austrian Science Fund, P24302-N18 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, K 125569 Funding text 1: Received by the editors May 7, 2017, and, in revised form, August 17, 2017. 2010 Mathematics Subject Classification. Primary 42A55; Secondary 42A61, 30B50, 11K38. The first author’s research was supported by FWF Grant P24302-N18 and NKFIH grant K 125569. Funding Agency and Grant Number: FWFAustrian Science Fund (FWF) [P24302-N18]; NKFIH [K 125569] Funding text: The first author's research was supported by FWF Grant P24302-N18 and NKFIH grant K 125569. Cited By :1 Export Date: 8 February 2020 Funding details: Austrian Science Fund, P24302-N18 Funding details: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, K 125569 Funding text 1: Received by the editors May 7, 2017, and, in revised form, August 17, 2017. 2010 Mathematics Subject Classification. Primary 42A55; Secondary 42A61, 30B50, 11K38. The first author’s research was supported by FWF Grant P24302-N18 and NKFIH grant K 125569. AB - The asymptotic behavior of exponential sums ΣN k=1 exp(2πinkα) for Hadamard lacunary (nk) is well known, but for general (nk) very few precise results exist, due to number theoretic difficulties. It is therefore natural to consider random (nk) and in this paper we prove the law of the iterated logarithm for ΣN k=1 exp(2πinkα) if the gaps nk+1 − nk are independent, identically distributed random variables. As a comparison, we give a lower bound for the discrepancy of {nkα} under the same random model, exhibiting a completely different behavior. LA - English DB - MTMT ER - TY - JOUR AU - Berkes, István TI - Strong approximation and a central limit theorem for St. Petersburg sums JF - STOCHASTIC PROCESSES AND THEIR APPLICATIONS J2 - STOCH PROC APPL VL - 129 PY - 2019 IS - 11 SP - 4500 EP - 4509 PG - 10 SN - 0304-4149 DO - 10.1016/j.spa.2018.12.003 UR - https://m2.mtmt.hu/api/publication/3414590 ID - 3414590 N1 - Accepted in August 2018; Available online 19 December 2018 Export Date: 8 January 2019 Article in Press CODEN: STOPB Export Date: 19 September 2019 Article in Press CODEN: STOPB Export Date: 8 February 2020 CODEN: STOPB AB - The St. Petersburg paradox (Bernoulli 1738) concerns the fair entry fee in a game where the winnings are distributed as P(X = 2k) = 2−k, k = 1, 2, . . .. The tails of X are not regularly varying and the sequence Sn of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-L¨of (1985), Cs¨org˝o and Dodunekova (1991)). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that Sn can be approximated by a semistable L´evy process {L(n), n ≥ 1} with a.s. error O( √ n(log n)1+ε) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory. LA - English DB - MTMT ER -