A note on the Davis-Gut law

作者：Liu Xiangdong^*; Guo Hui

来源：Statistics & Probability Letters, 2015, 105: 163-167.

DOI：10.1016/j.sp1.2015.06.018

• Davis-Gut law
• Law of the iterated logarithm

Let {X, X-n >= 1} be a sequence of independent and identically distributed random variables with EX = 0 and EX2 = 1. Let alpha(n) = (EXI)-I-2(vertical bar X vertical bar > root n log log n)/(EXI)-I-2(vertical bar X vertical bar <= root n log log n), n >= 1. In this note, a supplement to the Davis-Gut law is provided by proving that Sigma(infinity)(n=1) n(-1) {vertical bar Sigma(n)(k=1) X-k vertical bar > root 2n log log n} < infinity or = infinity according to Sigma(infinity)(n=1) (n log n root log log n)(-1) center dot (log n)(-alpha(n)) < infinity or = infinity. An example is given for illustrating the obtained result.

• 出版日期2015-10
• 单位暨南大学