arXiv subjects
Publications and source records attributed to Istvan Berkes.
In this paper, we extend the Marcinkiewicz--Zygmund inequality to the setting of Orlicz and Lorentz spaces. Furthermore, we generalize a Kadec--Pełczyński-type result -- originally established by the first and third authors for $L^p$ spaces with $1 \le p < 2$ -- to a broader class of Orlicz spaces defined via Young functions $ψ$ satisfying $x \le ψ(x) \le x^2$.
In a previous paper \cite{BT} we studied the asymptotic behavior of $\| \sum_{k=1}^N a_k X_{n_k}\|_p$ for lacunary sequences $(X_{n_k})$ of random variables in $L_p$ and used the result to give a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem in the case $1\le p<2$. In the present paper we extend this result for nonlinear functionals $f_k (a_1 X_{n_1}, \ldots, a_k X_{n_k})$, establishing a uniform version of the subsequence principle of Aldous \cite{ald}. Moreover, we prove Kadec-Pełczynski type theorems in Orlicz spaces $L_ψ$.
We show that every sequence $f_1, f_2, \cdots$ of real-valued random variables with $\sup_{n \in \N} \E (f_n^2) < \infty$ contains a subsequence $f_{k_1}, f_{k_2}, \cdots$ converging in \textsc{Cesàro} mean to some $\,f_\infty \in \mathbb{L}^2$ {\it completely,} to wit, $ \sum_{N \in \N} \, ¶\left( \bigg| \frac{1}{N} \sum_{n=1}^N f_{k_n} - f_\infty \bigg| > \eps \right)< \infty\,, \quad \forall ~ \eps > 0\,; $ and {\it hereditarily,} i.e., along all further subsequences as well. We also identify a condition, slightly weaker than boundedness in $ \mathbb{L}^2,$ which turns out to be not only sufficient for the above hereditary complete convergence in \textsc{Cesàro} mean, but necessary as well.
The celebrated theorem of Komlos asserts that L1-boundedness is sufficient for a given sequence of functions to contain a subsequence along which (in a "lacunary" manner), and along whose every further subsequence ("hereditarily"), a strong law of large numbers holds. We identify here slightly weaker, Egorov-type conditions, as not only sufficient in this context, but necessary as well. Necessary and sufficient conditions are developed also for the lacunary/hereditary version of the weak law of large numbers for general sequences, as well as for the weak law of large numbers in the context of exchangeable sequences, both long-open questions.
In this paper we present the theory of lacunary trigonometric sums and lacunary sums of dilated functions, from the origins of the subject up to recent developments. We describe the connections with mathematical topics such as equidistribution and discrepancy, metric number theory, normality, pseudorandomness, Diophantine equations, and the subsequence principle. In the final section of the paper we prove new results which provide necessary and sufficient conditions for the central limit theorem for subsequences, in the spirit of Nikishin's resonance theorem for convergence systems. More precisely, we characterize those sequences of random variables which allow to extract a subsequence satisfying a strong form of the central limit theorem.
Random walks on the circle group $\mathbb{R}/\mathbb{Z}$ whose elementary steps are lattice variables with span $α\not\in \mathbb{Q}$ or $p/q \in \mathbb{Q}$ taken mod $\mathbb{Z}$ exhibit delicate behavior. In the rational case we have a random walk on the finite cyclic subgroup $\mathbb{Z}_q$, 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 $α$, and explicitly describe the transition of these Markov chains from finite to general state space as $p/q \to α$ 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 surprising transition from polynomial to exponential decay after $\approx q^2$ 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.
Let $α$ be an irrational number, let $X_1, X_2, \ldots$ be independent, identically distributed, integer-valued random variables, and put $S_k=\sum_{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails $P (|X_1|>t)\sim ct^{-β}$, $0<β<2$, in Part I of this paper we proved that, up to logarithmic factors, the order of magnitude of the discrepancy $D_N (S_k α)$ of the first $N$ terms of the sequence $\{S_k α\}$ is $O(N^{-τ})$, where $τ= \min (1/(βγ), 1/2)$ (with $β=2$ in the case of finite variances) and $γ$ is the strong Diophantine type of $α$. This shows a change of behavior of the discrepancy at $βγ=2$. In this paper we determine the exact order of magnitude of $D_N (S_k α)$ for $βγ<1$, and determine the limit distribution of $N^{-1/2} D_N (S_k α)$. We also prove a functional version of these results describing the asymptotic behavior of a wide class of functionals of the sequence $\{S_k α\}$. Finally, we extend our results to the discrepancy of $\{S_k\}$ for general random walks $S_k$ without arithmetic conditions on $X_1$, assuming only a mild polynomial rate on the weak convergence of $\{S_k\}$ to the uniform distribution.
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 $\{n_k α\}$, with the exception of metric results for exponentially growing $(n_k)$. It is therefore natural to consider random $(n_k)$, and in this paper we give nearly optimal bounds for the discrepancy of $\{n_k α\}$ in the case when the gaps $n_{k+1}-n_k$ 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 $n_{k+1}-n_k$ and the rational approximation properties of $α$. We also point out an interesting critical phenomenon, a sudden change of the order of magnitude of the discrepancy of $\{n_k α\}$ as the Diophantine type of $α$ passes through a certain critical value.
By a classical result of Kadec and Pełczynski (1962), every normalized weakly null sequence in $L^p$, $p>2$ contains a subsequence equivalent to the unit vector basis of $\ell^2$ or to the unit vector basis of $\ell^p$. In this paper we investigate the case $1\le p<2$ and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczynski theorem is that the limit random measure $μ$ of the sequence satisfies $\int_{\mathbb{R}} x^2 dμ(x)\in L^{p/2}$.
We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal{O}(j^{-α})$ for some $α\in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.
Let $f: {\mathbb R}\to {\mathbb R}$ be a measurable function satisfying \begin{equation*} f(x+1)=f(x), \qquad \int_0^1 f(x)\, dx=0, \qquad \int_0^1 f^2(x)\, dx<\infty. \end{equation*} The asymptotic properties of series $\sum c_k f(kx)$ have been studied extensively in the literature and turned out to be, in general, quite different from those of the trigonometric system. As the theory shows, the behavior of such series is determined by a combination of analytic, probabilistic and number theoretic effects, resulting in highly interesting phenomena not encountered in classical harmonic analysis. In this paper we survey some recent results in the field and prove asymptotic results for the system $\{f(nx), n\ge 1\}$ in the case when the function $f$ is not square integrable.
It is known that for any smooth periodic function $f$ the sequence $(f(2^kx))_{k\ge 1}$ behaves like a sequence of i.i.d.\ random variables, for example, it satisfies the central limit theorem and the law of the iterated logarithm. Recently Fukuyama showed that permuting $(f(2^kx))_{k\ge 1}$ can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal condition on $(n_k)_{k\ge 1}$, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of the iterated logarithm for any permutation of the sequence $(f(n_k x))_{k \geq 1}$. A similar result is proved for the discrepancy of the sequence $(\{n_k x\})_{k \geq 1}$, where $\{ \cdot \}$ denotes fractional part.
It is a well known fact that for periodic measurable $f$ and rapidly increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a sequence of independent, identically distributed random variables. For example, if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies the central limit theorem, the law of the iterated logarithm and several further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence remains i.i.d.\ after any permutation of its terms, it is natural to expect that the asymptotic properties of lacunary series are also permutation-invariant. Recently, however, Fukuyama (2009) showed that a rearrangement of the sequence $(f(2^kx))_{k\ge 1}$ can change substantially its asymptotic behavior, a very surprising result. The purpose of the present paper is to investigate this interesting phenomenon in detail and to give necessary and sufficient criteria for the permutation-invariance of the CLT and LIL for $f(n_kx)$.
Let ${\cal H}=(q_1, \ldots q_r)$ be a finite set of coprime integers and let $n_1, n_2, \ldots$ denote the multiplicative semigroup generated by $\cal H$ and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory and they have remarkable probabilistic and ergodic properties. For example, the asymptotic properties of the sequence $\{n_kx\}$ are very similar to those of independent, identically distributed random variables; here $\{\cdot \}$ denotes fractional part. However, the behavior of this sequence depends sensitively on the generating elements of $(n_k)$ and the combination of probabilistic and number-theoretic effects results in a unique, highly interesting asymptotic behavior. In particular, the properties of $\{n_kx\}$ are not permutation invariant, in contrast to i.i.d. behavior. The purpose of this paper is to show that $\{n_kx\}$ satisfies a strong independence property ("interlaced mixing"), enabling one to determine the precise asymptotic behavior of permuted sums $S_N (σ)= \sum_{k=1}^N f(n_{σ(k)} x)$. As we will see, the behavior of $S_N(σ)$ still follows that of sums of independent random variables, but its growth speed (depending on $σ$) is given by the classical Gál function of Diophantine approximation theory. Some examples describing the class of possible growth functions are given.
By a classical principle of analysis, sufficiently thin subsequences of general sequences of functions behave like sequences of independent random variables. This observation not only explains the remarkable properties of lacunary trigonometric series, but also provides a powerful tool in many areas of analysis. In contrast to "true" random processes, however, the probabilistic structure of lacunary sequences is not permutation-invariant and the analytic properties of such sequences can change radically after rearrangement. The purpose of this paper is to survey some recent results of the authors on permuted function series. We will see that rearrangement properties of lacunary trigonometric series $\sum (a_k\cos n_kx+b_k \sin n_kx)$ and their nonharmonic analogues $\sum c_k f(n_kx)$ are intimately connected with the number theoretic properties of $(n_k)_{k \geq 1}$ and we will give a complete characterization of permutational invariance in terms of the Diophantine properties of $(n_k)_{k \geq 1}$. We will also see that in a certain statistical sense, permutational invariance is the "typical" behavior of lacunary sequences.
Let $f$ be a measurable function satisfying $$f(x+1)=f(x), \qquad \int_0^1 f(x) dx=0, \qquad \textrm{Var} ~f < + \infty,$$ and let $(n_k)_{k\ge 1}$ be a sequence of integers satisfying $n_{k+1}/n_k \ge q >1$ $(k=1, 2, \ldots)$. By the classical theory of lacunary series, under suitable Diophantine conditions on $n_k$, $(f(n_kx))_{k\ge 1}$ satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences $(n_k)_{k\ge 1}$ as well, but as Fukuyama (2009) showed, the behavior of $f(n_kx)$ is generally not permutation-invariant, e.g. a rearrangement of the sequence can ruin the CLT and LIL. In this paper we construct an infinite order Diophantine condition implying the permutation-invariant CLT and LIL without any growth conditions on $(n_k)_{k\ge 1}$ and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if $f(x)=\sin 2πx$ and $(n_k)_{k\ge 1}$ grows almost exponentially. Finally we prove that, in a suitable statistical sense, for almost all sequences $(n_k)_{k\ge 1}$ growing faster than polynomially, $(f(n_kx))_{k\ge 1}$ has permutation-invariant behavior.
Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2α}}{(n_k n_{\ell})^α}] are proved, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0<α\le 1$; the estimate for $α=1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $α=1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson--Hunt-type inequality for systems of dilated functions of bounded variation or belonging to $\lip12$, a result that in turn settles two longstanding problems on the a.e.\ behavior of systems of dilated functions: the a.e. growth of sums of the form $\sum_{k=1}^N f(n_k x)$ and the a.e.\ convergence of $\sum_{k=1}^\infty c_k f(n_kx)$ when $f$ is 1-periodic and of bounded variation or in $\lip12$.
We prove the optimality of a criterion of Koksma (1953) in Khinchin's conjecture, settling a long standing open problem in analysis. Using this result, we also give a near optimal condition for the a.e.\ convergence of series $\sum_{k=1}^\infty c_k f(kx)$ for $f\in L^2$.