Precise Asymptotics in the Baum–Katz and Davis Laws of Large Numbers

AbstractLet X,X1,X2,… be a sequence of i.i.d. random variables such that EX=0, let Z be a random variable possessing a stable distribution G with exponent α, 1<α≤2, assume the distribution of X is attracted to G, and set Sn=X1+···+Xn. We prove that∑n≥1nr/p−2P|Sn|≥εn1/p∼ε−(αp/(α−p))(r/p−1)pr−pE|Z|(αp/(α−p))(r/p−1)as ε↘0,for 1≤p<r<α, under the additional assumption that there is normal attraction to G, and that∑n≥11nP|Sn|≥εn1/p∼αpα−p−logεas ε↘0,for 1≤p