Logarithmic multifractal spectrum of stable occupation measure

For a stable subordinator Yt of index α, 0<α<1, the occupation measureμ(A)=|{t∈[0,1] : Yt∈A}|is known to have (with probability 1) the property thatlim infr↓0ln μ(x−r, x+r)ln r=α, ∀x∈Y[0,1].In order to obtain an interesting spectrum for the large values of μ(x−r,x+r), we consider the setBθ=x∈Y[0,1] : lim supr↓0μ(x−r,x+r)cαrα(ln(1/r))1−α=θ,where cα is a suitable constant. It is shown that Bθ=∅ for θ>1, and Bθ ≠ ∅ for 0⩽θ⩽1; moreover, dim Bθ=Dim Bθ=α(1−θ1/(1−α)).