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The convergence of \(F\)-variation for Gaussian random fields. (English. Ukrainian original) Zbl 0955.60035
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.
MSC:
60G15 Gaussian processes
60G60 Random fields
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