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

Abstract

Let 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 S_n = X₁ + ··· + X_n. We prove that$\text{∑}\text{n}\text{≥}\text{1}\text{n}{}^{\mathrm{<mtext>r</mtext><mtext>/</mtext><mtext>p</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>-</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>2</mtext>}}\text{P}\text{|}\text{S}{}_{\mathrm{n}}\text{|}\text{≥}\text{ε}\text{n}{}^{\mathrm{<mtext>1/</mtext><mtext>p</mtext>}}\text{∼}\text{ε}{}^{\mathrm{<mtext>-</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>(\alpha </mtext><mtext>p</mtext><mtext>/(\alpha </mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>-</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>p</mtext><mtext>))(</mtext><mtext>r</mtext><mtext>/</mtext><mtext>p</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>-</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>1)</mtext>}}\text{p}\text{r}\text{−}\text{p}\text{E|Z|}{}^{\mathrm{<mtext>(\alpha </mtext><mtext>p</mtext><mtext>/(\alpha </mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>-</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>p</mtext><mtext>))(</mtext><mtext>r</mtext><mtext>/</mtext><mtext>p</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>-</mtext><mspace\; xmlns="true"\; sp="0.2"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>1)</mtext>}}\text{as ε}\text{↘}\text{0,}$for 1 ≤ p < r < α, under the additional assumption that there is normal attraction to G, and that$\text{∑}\text{n}\text{≥}\text{1}\text{1}\text{n}\text{P}\text{|}\text{S}{}_{\mathrm{n}}\text{|}\text{≥}\text{ε}\text{n}{}^{\mathrm{<mtext>1/</mtext><mtext>p</mtext>}}\text{∼}\text{α}\text{p}\text{α}\text{−}\text{p}\text{−}\text{log}\text{ε}\text{as ε}\text{↘}\text{0,}$for 1 ≤ p < α.

We close with a related result for the limiting case α = r = p = 2.