3.17 Let V be an open bounded region in d. If {X(t)} is a stationary random field with covariance function ????(h), define the (continuous) sample

3.17 Let V be an open bounded region in ℝd. If {X(t)} is a stationary random field with covariance function ????(h), define the (continuous) sample mean within V by X(V) = |V|

−1

∫V X(t) dt.

Show that its variance is given by var{X(V)} = |V|

−2

∫V ∫V

????(s − t) ds dt.

By expanding the square for the sample continuous variance within V in the formula for the dispersion variance

????(0|V) = |V|

−1 E

{

∫ [X(t) − X(V)]2 dt}

, show that

????(0|V) = ????(0) − var{X(V)}.