A note on the Davis–Gut law

Let {X,Xn≥1}{X,Xn≥1} be a sequence of independent and identically distributed random variables with EX=0EX=0 and EX2=1EX2=1. Let α(n)=EX2I(|X|>nloglogn)/EX2I(|X|≤nloglogn), n≥1n≥1. In this note, a supplement to the Davis–Gut law is provided by proving that ∑n=1∞n−1{|∑k=1nXk|>2nloglogn}<∞ or =∞=∞ according to ∑n=1∞(nlognloglogn)−1⋅(logn)−α(n)<∞ or =∞=∞. An example is given for illustrating the obtained result.