Large deviation principles for lacunary sums
Large deviation principles for lacunary sums
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缺额金额大偏差原则
DOI：
10.1090/tran/8788
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发表时间：
2022
影响因子：
1.3
通讯作者：
Ramanan, Kavita
中科院分区：
文献类型：
--
作者：
Aistleitner, Christoph;Gantert, Nina;Kabluchko, Zakhar;Prochno, Joscha;Ramanan, Kavita
设为满足Hadamard间隙条件的递增正整数序列，令\开始{方程 *} S_n（\omega）=\sum _ {k= 1}^ n\cos（2\pi a_k\omega），\qquad n\in\mathbb N，\;\omega\in [0，1].\这称为缺项三角和，可以看作是定义在概率空间上的一个随机变量，赋予它Lebesgue测度.已知缺项和具有独立随机变量和的几个典型性质。例如，一个中心极限定理已获得塞勒姆和Zygmund，而法律的重对数是由于埃尔德什和加尔。本文研究缺项和的大偏差原理。具体地说，在大间隙条件下，我们证明了序列确实满足大偏差原理，其速度和速率函数与反正弦分布的独立随机变量和相同。另一方面，我们表明，大偏差原则可能无法举行时，我们只假设阿达玛间隙条件。然而，我们表明，在特殊情况下，当对于一些，满足一个大偏差的原则（与速度）和率函数是不同的，并描述了一种算法来计算任意数量的条款中的泰勒展开。此外，我们还证明了它逐点收敛于α。此外，我们构造了一个随机扰动的序列，对于它，但同时满足一个大偏差原理，具有与独立情况下相同的速率函数，这是令人惊讶的不同的速率函数，人们可能天真地期望。我们将这一事实与某些丢番图方程的解的个数联系起来。总之，这些结果表明缺项三角和的大偏差原理对序列的算术性质非常敏感。这是特别值得注意的，因为没有这样的算术效果是可见的中心极限定理或在法律的重对数缺三角和。我们的证明使用的工具，从概率论，谐波分析和动力系统的组合。引用
Letbe an increasing sequence of positive integers satisfying the Hadamard gap conditionfor all, and let\begin {equation*} S_n (\omega)=\sum _ {k= 1}^ n\cos (2\pi a_k\omega),\qquad n\in\mathbb N,\;\omega\in [0, 1].\end {equation*} Thenis called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability spaceendowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem forhas been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition, we prove that the sequencedoes indeed satisfy a large deviation principle with speedand the same rate functionas for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case whenfor some,satisfies a large deviation principle (with speed) and a rate functionthat is different from, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of. In addition, we also prove thatconverges pointwise toas. Furthermore, we construct a random perturbationof the sequencefor whichas, but for which at the same timesatisfies a large deviation principle with the same rate functionas in the independent case, which is surprisingly different from the rate functionone might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem or in the law of the iterated logarithm for lacunary trigonometric sums. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems. References