Sharp near-unit persistence constant for symmetric stable AR(1) sequences

Determine whether, for every stability index 0 < α < 2, the persistence exponent Λ(a, α) of a symmetric α-stable autoregressive sequence satisfies lim_{a ↑ 1} Λ(a, α)/log(1/a) = α/2; equivalently, establish whether the dense-sampling persistence exponent λα(h) of the stationary symmetric α-stable Lamperti process satisfies lim_{h ↓ 0} λα(h)/h = 1/2.

Background

The paper studies the exponential persistence rate Λ(a, α) for the symmetric α-stable AR(1) recursion X_n = aX_{n−1} + ξ_n as the autoregressive coefficient a approaches one. Its main bounds prove that Λ(a, α) is of order log(1/a) and that the normalized rate converges to a constant in (0, α/2]. The value α/2 is known in the Gaussian case α = 2 but remains unresolved for 0 < α < 2.

Through a Lamperti transformation, the near-unit-root limit is equivalent to a dense-sampling problem for a stationary symmetric α-stable Ornstein–Uhlenbeck-type process. The unresolved issue is whether a discretely observed jump process has the same first-order persistence rate as continuous monitoring. A sampled path can cross below zero and return above zero between observations, and controlling these rescued sub-mesh crossings is the principal obstacle identified by the paper.