论文标题
大偏差原则
Large Deviation Principles for Lacunary Sums
论文作者
论文摘要
令$（a_k）_ {k \ in \ mathbb n} $是满足Hadamard Gap条件$ a_ {k+1}/a_k> q> 1 $ in \ in \ mathbb n $的整数序列ω），\ qquad n \ in \ mathbb n，\;ω\ in [0,1]。 $$已知lacunary Trigonometric sum $ s_n $可以展示几种独立随机变量总和的典型属性。在本文中，我们以$ s_n $的价格启动对大偏差原理（LDP）的调查。在较大的间隙条件下$ a_ {k+1}/a_k \ to \ to \ infty $，我们证明$（s_n/n）_ {n \ in \ Mathbb n} $满足LDP的速度$ n $，并且同样的速率函数$ \ tilde {i} $仅适用于独立范围，但要符合独立的变量。假设Hadamard间隙条件。但是，我们证明，在特殊情况下，$ a_k = q^k $对于某些$ q \ in \ {2,3，\ ldots \} $，$（s_n/n）_ {n \ in \ mathbb n} $满足$ n $的LDP，并满足$ n $的$ i_q $ $ i_q $ $ i_q $ $ i_q $与$ \ \ tilde} $ {i}。我们还表明，$ i_q $将$ \ tilde i $作为$ q \ to \ to \ infty $收敛，并构建一个随机的扰动$（a_k）_ {k \ in \ Mathbb n} $的序列$（2^K） $ k \ to \ infty $，但对于$（s_n/n）_ {n \ in \ mathbb n} $，与速率函数$ \ tilde {i} $相满于独立案例，而不是独立的情况，而不是这样，因为一个人可能会带来na { ^} vely vely，并带有利率函数$ i_2 $ $ i_2 $。我们将这一事实与某些二磷剂方程的解决方案的数量联系起来。我们的结果表明，lacunary Trigonometric Sums的LDP对$（a_k）_ {k \ in \ Mathbb n} $的算术属性敏感。这尤其值得注意，因为Salem和Zygmund或Erdös和Gál的迭代对数定律中，在中心限制定理中看不到这种算术效应。我们的证明使用了概率理论，谐波分析和动态系统的工具组合。
Let $(a_k)_{k\in\mathbb N}$ be a sequence of integers satisfying the Hadamard gap condition $a_{k+1}/a_k>q>1$ for all $k\in\mathbb N$, and let $$ S_n(ω) = \sum_{k=1}^n\cos(2πa_k ω),\qquad n\in\mathbb N,\;ω\in [0,1]. $$ The lacunary trigonometric sum $S_n$ is known to exhibit several properties typical for sums of independent random variables. In this paper we initiate the investigation of large deviation principles (LDPs) for $S_n$. Under the large gap condition $a_{k+1}/a_k\to\infty$, we prove that $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with speed $n$ and the same rate function $\tilde{I}$ as for sums of independent random variables with the arcsine distribution, but show that the LDP may fail to hold when we only assume the Hadamard gap condition. However, we prove that in the special case $a_k=q^k$ for some $q\in \{2,3,\ldots\}$, $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with speed $n$ and a rate function $I_q$ different from $\tilde{I}$. We also show that $I_q$ converges pointwise to $\tilde I$ as $q\to\infty$ and construct a random perturbation $(a_k)_{k\in\mathbb N}$ of the sequence $(2^k)_{k\in\mathbb N}$ for which $a_{k+1}/a_k\to 2$ as $k\to\infty$, but for which $(S_n/n)_{n\in\mathbb N}$ satisfies an LDP with the rate function $\tilde{I}$ as in the independent case and not, as one might na{ï}vely expect, with rate function $I_2$. We relate this fact to the number of solutions of certain Diophantine equations. Our results show that LDPs for lacunary trigonometric sums are sensitive to the arithmetic properties of $(a_k)_{k\in\mathbb N}$. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem by Salem and Zygmund or in the law of the iterated logarithm by Erdös and Gál. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems.
