Central limit theorems for the L 2 norm of increments of local times of Lévy processes ∗

Let X = {Xt, t ∈ R+} be a symmetric Lévy process with local time {Lt ; (x, t) ∈ R × R +}. When the Lévy exponent ψ(λ) is regularly varying at zero with index 1 < β ≤ 2, and satisfies some additional regularity conditions, lim t→∞ ∫∞ −∞(L x+1 t − Lt ) dx− E (∫∞ −∞(L x+1 t − Lt ) dx ) t √ ψ−1(1/t) L = (8cψ,1) 1/2 (∫ ∞ −∞ ( Lβ,1 )2 dx )1/2 η, where Lβ,1 = {Lβ,1 ; x ∈ R} denotes the local time, at time 1, of a symmetric stable process with index β, η is a normal random variable with mean zero and variance one that is independent of Lβ,1, and cψ,1 is a known constant that depends on ψ. When the Lévy exponent ψ(λ) is regularly varying at infinity with index 1 < β ≤ 2 and satisfies some additional regularity conditions lim h→0 √ hψ2(1/h) {∫ ∞ −∞ (L 1 − L x 1) 2 dx− E (∫ ∞ −∞ (L 1 − L x 1) 2 dx )} L = (8cβ,1) 1/2 η (∫ ∞ −∞ (L1) 2 dx )1/2 , where η is a normal random variable with mean zero and variance one that is independent of {L1 , x ∈ R}, and cβ,1 is a known constant.