The functional law of the iterated logarithm for Lévy's area process

Abstract

Let {1/2 L(t):t≥0} be Lévy's area process and define a random sequence {f _n :n≥1} in C[0,1] by f _n (t)=L(nt)/nl(n), tε[0,1], n≥1, where l(n)=1 for n-1,2 and l(n)=loglog n for n≥3. It is shown that, with probability 1, the set of limit points of {f _n :n≥1} is H={x:x(t)=∫ ^t₀ (z ₁(u)ż₁(u)z ₂(u))du, (z ₁,z(₂)εK, tε[0,1]}, where K-{(z ₁,z ₂):z _j (0)=0, z _j absolutely continuous, j=1,2, 1/2 ∫ ¹₀ (|ż₁(u)|²+|ż₂(u)|² du≤1}.

Work supported by the Natural sciences and Engineering Research Council of Canada, by the Fonds F.C.A.R. of the Province of Quebec, and by the Deutsche Forschungsgmeinschaft.