3.1 Let ????(h), h d, be a valid semivariogram and suppose it tends to a finite...

3.1 Let ????(h), h ∈ ℝd, be a valid semivariogram and suppose it tends to a finite limit for large lags, ????(h) → c as |h| → ∞. Show that

????(h) = c − ????(h)

defines a valid covariance function ????(h) with marginal variance

????(0) = c.

Hint: Let {XI(t)} denote an intrinsic process with semivariogram ????(h) and with vanishing incremental mean. Let D denote a fixed bounded open set in

ℝd (such as a disk) with volume |D|, and let ???? > 0 be a dilation parameter.

Define a (nonstationary) process {Y(t)} by Y(t) = XI(t) − 1

|????D|∫????D XI(s)ds.