Abstract.
Consider 0<α<1 and the Gaussian process Y(t) on ℝN with covariance E(Y(s)Y(t))=|t|2α+|s|2α−|t−s|2α, where |t| is the Euclidean norm of t. Consider independent copies X 1,…,X d of Y and␣the process X(t)=(X 1(t),…,X d(t)) valued in ℝd. When kN≤␣(k−1)αd, we show that the trajectories of X do not have k-multiple points. If N<αd and kN>(k−1)αd, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛk N /α−( k −1) d (loglog(1/ɛ))k. If N=αd, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function ϕ(ɛ)=ɛd(log(1/ɛ) logloglog 1/ɛ)k. (This includes the case k=1.)
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Received: 20 May 1997 / Revised version: 15 May 1998
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Talagrand, M. Multiple points of trajectories of multiparameter fractional Brownian motion. Probab Theory Relat Fields 112, 545–563 (1998). https://doi.org/10.1007/s004400050200
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DOI: https://doi.org/10.1007/s004400050200