Edgeworth expansion for ergodic diffusions
The present paper validates the Edgeworth expansion for continuous-time strong Markov processes. Let \(X = (X_t )_{t \geq 0}\) be such a process and \(Z = (Z_t )_{t \geq 0}\) be an \(n\)-additive functional of \(X\). Denote by \(P_\nu \) the law of \(X\) with initial distribution \(\nu\). It is proved that for a given sufficiently smooth function \(A:\mathbb{R}^n \to \mathbb{R}^1\), \[ \sup_B \left| {P_\nu \left[ {\sqrt T (A(Z_T /T) - A(\gamma ))/\sigma \in B} \right] - \int_B {p_T (z)dz} } \right| = O(T^{ - 1})\quad\text{as }T \to \infty, \] where supremum is taken over all Borel sets, \[ p_T(z) = \varphi(z)\left\{ 1 + T^{ - 1/2} \left\{ A_1 z/\sigma + A_2 (z^3 - 3z)/6\sigma ^3 \right\} \right\}, \] \(\varphi\) is the density of the standard normal distribution and \(\gamma \), \(\sigma \), \(A_j \), j=1,2 are constants written explicitly in terms of moments of certain distributions depending on \(X\) and \(Z\). In particular for diffusions of dimension one, verifiable conditions for the validity of the expansion are given on terms of the coefficients of the corresponding stochastic differential equations.
scientific article; zbMATH DE number 5351821
scientific article