6.12 Verify that the family of distributions defined in (3.18) satisfies con- ditions (3.1) and (3.2) of Theorem 6.3.1. (Hint: Use the fact that for any & 2 1, any A = (#1, /42. ...; () ERk, and any nonnegative definite & x & matrix _ = ((0;;))xxx, there is a unique probability distribution & such that for any s = ($1, $2, . .., $%) in RK, exp Silk v(dr) 1=1 k k = exp E Silli + 1=1 i=l j=1 Observe that this implies that for s = ($1, $2. .... s,) in R*, the in- duced distribution (under v) on R by the map g(@) = >in air from R* - R is univariate normal with mean >iz, sip; and varianceKolmogorov's Probability Model Random Variables and Random Vectors Kolmogorov's Consistency Theorem ooo00000000000000000 ooooooooo0000000000000008000.00000 Kolmogorov's Consistency Theorem Example 6.3.9. (Gaussian processes). Let A be a non-empty set and {Xa : a E A} be a stochastic process. Such a process is called Gaussian if for {a1, . .., OK} C A and real numbers t1, . . . , tk, the K random variable _ tXox has a univariate normal distribution k=1 (with possibly zero variance). For such a process, the functions u(a) = EXa and o(a, a') = Cov (Xa, Xo') are called the mean K and covariance functions, respectively. Since Var _ tk Xak) 2 0, *=1 it follows that for any t1, . . . , tK, K K EE titjo (ai, a;) > 0. (3.17) i=1 j=1 This property of the covariance function o(., .) is called non-negative definiteness.Kolmogorov's Probability Model Random Variables and Random Vectors Kolmogorov's Consistency Theorem ooooooooo00000000000 oooooooooooooooooo00000008000.0000 Kolmogorov's Consistency Theorem A natural question is: Given functions / : A - R and o : A x A -> R such that o is symmetric and satisfies (3.17), does there exist a Gaussian process {Xa : a E A} with H(.) and o(., .) as its mean and covariance functions, respectively? The answer is yes and it follows from Theorem 6.3.1 by defining the family QA of finite dimensional distributions as follows. Let V(a1,...,ak >sisjo ( Qi, oj ) (3.18) 1= 1 1=1 /=1 for $1, . . . , SK in R. If the matrix E = ((o (a;, a;) )), 1