5.2 This exercise looks at the regularity of the Matrn covariance function for small lags. This behavior

5.2 This exercise looks at the regularity of the Matérn covariance function for small lags. This behavior is important for the study of infill asymptotics in Section 5.14. Suppose the real index ???? is not a negative integer and let z be a positive number. The modified Bessel function of the first kind I???? (z) has the limiting behavior I???? (z) = (z∕2)

????

Γ(1 + ????)

{

1 + O(z2

)

}

as z → 0 for fixed ???? (e.g. Abramowitz and Stegun, 1964, p. 375). The modified Bessel function of the second kind K???? (z) is defined by K???? (z) = ????∕2 sin ????????

{

I−???? (z) − I???? (z)

}

.

For 0

Γ(????)

(z∕2)

????K???? (z) = 1 − Γ(1 − ????)

Γ(1 + ????)

(z∕2)

2???? + O(z2

)

as z → 0. Hence, for 0