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Yander Wiel

Appendix Proofs of Theorems 1 through 3

Proof of Theorem 1    Clearly Z = (Z₀,...,Z,) is multinormal with each

Z_k~N(0,1). Let, 9 = 1 - y, a_k = l/Vl-0²⁽*⁺¹⁾ and for m < k let =e*-'"(l-e^2{”^+l)). Then, for m<,k,

=co\{Z_m,Z_k} = a_ma_kb_mk and for j<m<k,

co\{Zj,Z_k\Z_m] = a_Jk -ct^W⁰-

⁼ {bjkbmm ~ brnk^jm )^aj^ak

But this conditional covariance is 0 because

Thus, Z₀,..., Z, is Markovian. Because -Z~Z the Lemma below implies that |Z₀|,...,|Z,| is also Markovian. Theorem 3 defines g*(iv) as the density of |Z*_,| conditional on \Z_k\ = v. It is straightforward to show that this density is

8k W v) = <t»(H    1 “'^CT*-i jk) + 4>(!" w*.,,* ,1 -    )

where <J>('|p, a²) is the N(p, a²) density.

Lemma 1 Jf the elements ofX= (X_l,...,X_n) form a Markov chain with joint density f[\) such that J(x) = J[-x), then the elements of Y = (K, = \X_t |,..., Y_n = \X_n\) also form a Markov chain.

Proof

For i < j let f^ify) denote the density of X_y = \x_i,...,Xand let denote the density of Y_y.    7,). Because the elements of X are