@article{MTMT:33670168,
	title = {Some optimal conditions for the ASCLT},
	url = {https://m2.mtmt.hu/api/publication/33670168},
	author = {Berkes, István and Siegfried, Hörmann},
	doi = {10.1007/s10959-023-01245-w},
	journal-iso = {J THEOR PROBAB},
	journal = {JOURNAL OF THEORETICAL PROBABILITY},
	volume = {37},
	unique-id = {33670168},
	issn = {0894-9840},
	year = {2024},
	eissn = {1572-9230},
	pages = {209-227}
}

@article{MTMT:33670095,
	title = {Lacunary sequences in analysis, probability and number theory},
	url = {https://m2.mtmt.hu/api/publication/33670095},
	author = {Christoph, Aistleitner and Berkes, István and Robert, Tichy},
	journal-iso = {LECT NOTES MATH},
	journal = {LECTURE NOTES IN MATHEMATICS},
	volume = {accepted},
	unique-id = {33670095},
	issn = {0075-8434},
	year = {2024},
	eissn = {1617-9692},
	pages = {&}
}

@article{MTMT:34236327,
	title = {Random walks on the circle and Diophantine approximation},
	url = {https://m2.mtmt.hu/api/publication/34236327},
	author = {Berkes, István and Borda, Bence},
	doi = {10.1112/jlms.12749},
	journal-iso = {J LOND MATH SOC},
	journal = {JOURNAL OF THE LONDON MATHEMATICAL SOCIETY},
	volume = {108},
	unique-id = {34236327},
	issn = {0024-6107},
	abstract = {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.},
	year = {2023},
	eissn = {1469-7750},
	pages = {409-440},
	orcid-numbers = {Borda, Bence/0000-0002-9066-8850}
}

@article{MTMT:34845921,
	title = {Trimmed Least Square Estimators for Stable Ar(1) Processes},
	url = {https://m2.mtmt.hu/api/publication/34845921},
	author = {Bazarova, Alina and Berkes, István and Horváth, Lajos},
	doi = {10.1556/314.2022.00003},
	journal-iso = {MATH PANNONICA},
	journal = {MATHEMATICA PANNONICA},
	volume = {28_NS2},
	unique-id = {34845921},
	issn = {0865-2090},
	abstract = {We prove the weak consistency of the trimmed least square estimator of the covariance parameter of an AR(1) process with stable errors.},
	year = {2022},
	pages = {16-23}
}

@article{MTMT:34845912,
	title = {On the Almost Sure Central Limit Theorem Along Subsequences},
	url = {https://m2.mtmt.hu/api/publication/34845912},
	author = {Berkes, István and Csáki, Endre},
	doi = {10.1556/314.2022.00002},
	journal-iso = {MATH PANNONICA},
	journal = {MATHEMATICA PANNONICA},
	volume = {28_NS2},
	unique-id = {34845912},
	issn = {0865-2090},
	abstract = {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.},
	year = {2022},
	pages = {11-15}
}

@article{MTMT:32379817,
	title = {On the discrepancy of random subsequences of [n alpha}, II},
	url = {https://m2.mtmt.hu/api/publication/32379817},
	author = {Berkes, István and Borda, Bence},
	doi = {10.4064/aa200811-25-1},
	journal-iso = {ACTA ARITH},
	journal = {ACTA ARITHMETICA},
	volume = {199},
	unique-id = {32379817},
	issn = {0065-1036},
	keywords = {Random walk; Diophantine approximation; Discrepancy; Functional limit theorems; Nonstationary process},
	year = {2021},
	eissn = {1730-6264},
	pages = {303-330},
	orcid-numbers = {Borda, Bence/0000-0002-9066-8850}
}

@article{MTMT:30945205,
	title = {On the discrepancy of random subsequences of {n alpha}},
	url = {https://m2.mtmt.hu/api/publication/30945205},
	author = {Berkes, István and Borda, Bence},
	doi = {10.4064/aa180417-12-12},
	journal-iso = {ACTA ARITH},
	journal = {ACTA ARITHMETICA},
	volume = {191},
	unique-id = {30945205},
	issn = {0065-1036},
	abstract = {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.},
	keywords = {continued fractions; Random walk; Diophantine approximation; Discrepancy; critical phenomena},
	year = {2019},
	eissn = {1730-6264},
	pages = {383-415}
}

@article{MTMT:30803174,
	title = {On the discrepancy of random walks on the circle},
	url = {https://m2.mtmt.hu/api/publication/30803174},
	author = {Alina, Bazarova and Berkes, István and Marko, Raseta},
	doi = {10.2478/udt-2019-0015},
	journal-iso = {UNIF DISTRIB THEOR},
	journal = {UNIFORM DISTRIBUTION THEORY},
	volume = {14},
	unique-id = {30803174},
	issn = {1336-913X},
	year = {2019},
	eissn = {2309-5377},
	pages = {73-86}
}

@article{MTMT:3414600,
	title = {On the law of the iterated logarithm for random exponential sums},
	url = {https://m2.mtmt.hu/api/publication/3414600},
	author = {Berkes, István and Borda, Bence},
	doi = {10.1090/tran/7415},
	journal-iso = {T AM MATH SOC},
	journal = {TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY},
	volume = {371},
	unique-id = {3414600},
	issn = {0002-9947},
	abstract = {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.},
	year = {2019},
	eissn = {1088-6850},
	pages = {3259-3280}
}

@article{MTMT:3414590,
	title = {Strong approximation and a central limit theorem for St. Petersburg sums},
	url = {https://m2.mtmt.hu/api/publication/3414590},
	author = {Berkes, István},
	doi = {10.1016/j.spa.2018.12.003},
	journal-iso = {STOCH PROC APPL},
	journal = {STOCHASTIC PROCESSES AND THEIR APPLICATIONS},
	volume = {129},
	unique-id = {3414590},
	issn = {0304-4149},
	abstract = {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.},
	year = {2019},
	eissn = {1879-209X},
	pages = {4500-4509}
}