The convergence of \(F\)-variation for Gaussian random fields. (English. Ukrainian original) Zbl 0955.60035

Kurchenko, O. O.

Theory Probab. Math. Stat. 60, 113-123 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 102-111 (1999).

For any \(F: R\to R\) and a random field \(\xi(t)\), \(t\in[0,1]^d,\) the F-variation is defined as \[ V_n(F)=a_n^{-d}\sum F((\triangle \xi_k/E\triangle\xi_k)^2)^{1/2}, \] where \(\triangle\xi_k\) are increments of the field \(\xi\) on a uniform partitioning of \([0,1]^d.\) The paper deals with the Baxter-type theorems for \(V_n(F)\). Conditions are presented under which \(V_n(F)\) converges to some nonrandom constant \(C\not=0\) as \(n\to\infty\). A Chentsov field and a multiparameter fractional Brownian motion are considered as examples.

Reviewer: R.E.Maiboroda (Kyïv)

MSC:

60G15	Gaussian processes
60G60	Random fields

Keywords:

Baxter theorem; random field; Chentsov field; fractional Brownian motion