2.6 For t d let ????(t) = I[|t| < 1 2 ] where I is an indicator function. Let ????(h) = (???? ????

2.6 For t ∈ ℝd let ????(t) = I[|t| < 1 2

] where I is an indicator function. Let ????(h) =

(???? ∗ ????̌ )(h) as in Eq. (2.33). Show that ????#(r) = ????(|h|) takes the following form in dimensions d = 1,2, 3:

(d = 1) ????(r)∝(1 − r)I[r ≤ 1],

(d = 2) ????(r)∝{1 − (2∕????)r(1 − r2

)

1∕2 − (2∕????)sin−1 r}I[r < 1],

(d = 3) ????(r) ∝ {

1 − 3 2

r +

1 2

r3

}

I[r < 1].

The function for d = 3 is the “spherical scheme” of Table 2.1. The function for d = 2 is the analogous construction in two dimensions and is sometimes known as the “circular scheme.” The function for d = 1 is sometimes known as the “tent” scheme, since, when plotted as a function of h, with r = |h|, it has a tent-like shape between h = −1 and h = 1 with a peak at h = 0. Note that the tent scheme can also be viewed as a special case of the restricted-power scheme in Table 2.1 with k = 1, but it is a valid covariance function only in dimension d = 1. Chilés and Delfiner (2012, pp. 85–88)

discuss higher dimensional versions.

Hint: For d ≥ 2, ????#(r) is given by the volume in the intersection of two spheres of radius 1 2 and distance r apart. Since the volume of a

(d − 1)-dimensional sphere of radius r equals {????

1 2 (d−1)

∕Γ( 1 2

d + 1 2

)}rd−1 =

????d−1rd−1∕(d − 1) (to verify this formula, differentiate with respect to r at r = 1 to get the surface area ????d−1 of the sphere), ????#(r) can be expressed as