Ge Lin, Sang Hailin, Shao Qi-Man
Division of Arts and Sciences, Mississippi State University at Meridian, Meridian, MS 39307, USA.
Department of Mathematics, University of Mississippi, University, MS 38677, USA.
Entropy (Basel). 2025 Jan 7;27(1):41. doi: 10.3390/e27010041.
In this paper, we study self-normalized moderate deviations for degenerate -statistics of order 2. Let {Xi,i≥1} be i.i.d. random variables and consider symmetric and degenerate kernel functions in the form h(x,y)=∑l=1∞λlgl(x)gl(y), where λl>0, Egl(X1)=0, and gl(X1) is in the domain of attraction of a normal law for all l≥1. Under the condition ∑l=1∞λl<∞ and some truncated conditions for {gl(X1):l≥1}, we show that logP(∑1≤i≠j≤nh(Xi,Xj)max1≤l<∞λlVn,l2≥xn2)∼-xn22 for xn→∞ and xn=o(n), where Vn,l2=∑i=1ngl2(Xi). As application, a law of the iterated logarithm is also obtained.
在本文中，我们研究二阶退化统计量的自归一化适度偏差。设{Xi, i≥1}为独立同分布的随机变量，并考虑形式为h(x, y)=∑l = 1∞λlgl(x)gl(y)的对称且退化的核函数，其中λl>0，Egl(X1)=0，且对于所有l≥1，gl(X1)属于正态分布的吸引域。在条件∑l = 1∞λl<∞以及{gl(X1): l≥1}的一些截断条件下，我们证明当xn→∞且xn = o(n)时，logP(∑1≤i≠j≤nh(Xi, Xj) / max1≤l<∞λlVn, l2≥xn2)∼ - xn2 / 2，其中Vn, l2 = ∑i = 1ngl2(Xi)。作为应用，还得到了一个重对数律。
