论文标题
关于$ \ {nα\} $ ii的随机子序列的差异
On the discrepancy of random subsequences of $\{nα\}$ II
论文作者
论文摘要
令$α$为一个不合理的数字，令$ x_1，x_2，\ ldots $是独立的，相同分布，整数值随机变量，然后放置$ s_k = \ sum_ {j = 1}^k x_j $。假设$ x_1 $具有有限的差异或重尾$ P（| x_1 |> t）\ sim ct^{ - β} $，$ 0 <β<2 $，在本文的第一部分中，我们证明，我们证明了对数因素，差异$d_kα$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n（s_kk）$ n $ n $ n $。 α\} $是$ o（n^{ - τ}）$，其中$τ= \ min（1/（βγ），1/2），1/2）$（在有限方差的情况下$β= 2 $），$γ$是强的Dioophantine型$α$。这显示了$βγ= 2 $的差异行为的变化。在本文中，我们确定$ d_n（s_kα）$ $βγ<1 $的确切数量级，并确定$ n^{ - 1/2} d_n（s_kα）$的极限分布。我们还证明了这些结果的功能版本，描述了序列$ \ {s_kα\} $的广泛功能的渐近行为。最后，我们将结果扩展到$ \ {s_k \} $的一般随机步行$ s_k $的$ \ {s_k \} $，而没有算术条件，$ x_1 $，假设$ \ {s_k \} $的弱收敛性较小的多项式速率。
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.
