Parametrization Total variation merging

6.1 Parametrization

Let V = {0, 1} be the two-point space. Let 0 ≤ p ≤ 1, and set M p x, y = ¨ p if y = 0 1 − p if y = 1. 1 Obviously, this chain has stationary measure ν p z = ¨ p if x = 0 1 − p if x = 1. In fact, for any probability measure µ, µM p = ν p . Let I denote the 2× 2 identity matrix and consider the family of kernels M p = K[ α, p] : K[α, p] = 1 1 + α € αI + M p Š : α ∈ − min{p, 1 − p}, ∞ . By convention we have that I = K[∞, p] for all 0 ≤ p ≤ 1. Any kernel in M p has invariant measure ν p and M p is exactly the set of all Markov kernels on V with invariant measure ν p . Indeed, if q 1 , q 2 ∈ [0, 1] and K = ‚ q 1 1 − q 1 q 2 1 − q 2 Œ then solving the equation 1 + αK = αI + M p yields α = q 1 − q 2 1 + q 2 − q 1 and p = q 2 1 + q 2 − q 1 . Note that K[ α, p] has no holding in at least one point in V when α = − min{p, 1 − p}. Also, the only non irreducible kernels are I and those with min{p, 1 − p} = 0 whereas the only irreducible periodic kernel is K[−1 2, 12]. For later purpose, we note that the nontrivial eigenvalue β 1 K[α, p] of K[ α, p] is given by β 1 K[α, p] = α 1 + α . 2

6.2 Total variation merging

Let K i ∞ 1 with K i = K[α i , p i ], p i ∈ [0, 1], α i ∈ [− min{p i , 1 − p i }, ∞ ⊂ [−12, ∞. The following statement identifies K 0,n using the α, p-parametrization. Lemma 6.1. K 0,n = ‚ 1 1 + α 0,n Œ α 0,n I + M p 0,n = K[α 0,n , p 0,n ] where α 0,n = Q n i=1 α i Q n i=1 1 + α i − Q n i=1 α i 3 and p 0,n = P n i=1 Q i−1 j=1 1 + α j Q n j=i+1 α j p i Q n i=1 1 + α i − Q n i=1 α i . 4 1485 Proof. This can be shown by induction. Note that K 0,2 = K 1 K 2 is equal to 1 1 + α 0,2 α 0,2 I + M p 0,2 with α 0,2 = α 1 α 2 1 + α 1 + α 2 , p 0,2 = α 2 p 1 + 1 + α 1 p 2 1 + α 1 + α 2 . Using the two step formula above for K 0,n+1 = K 0,n K n+1 , we have α 0,n+1 = α 0,n α n+1 1 + α 0,n + α n+1 and p 0,n+1 = α n+1 p 0,n + 1 + α 0,n p n+1 1 + α 0,n + α n+1 . 5 To see that α 0,n+1 and p 0,n+1 can be written in the forms of 3 and 4, we use the induction hypothesis along with the equality 1 + α 0,n + α n+1 = Q n+1 i=1 1 + α i − Q n+1 i=1 α i Q n i=1 1 + α i − Q n i=1 α i . To study the merging properties of K i ∞ 1 , observe that, for any two initial probability measures µ and ν on V , we have µ n − ν n = µ K 0,n − ν K 0,n = ‚ 1 1 + α 0,n Œ α 0,n µ − ν I + µ − ν M p 0,n = ‚ α 0,n 1 + α 0,n Œ µ − ν = n Y i=1 α i 1 + α i µ − ν . 6 This, together with elementary considerations, yields the following proposition. Proposition 6.2. The sequence K i ∞ 1 is not merging in total variation if and only if the following three conditions hold: 1. For all i, α i 6= 0; 2. P i: α i 1 + 2α i ∞; 3. P i: α i 1 + α i −1 ∞. Remark 6.3. The meaning of this proposition is that, in order to avoid merging we must prevent α i = 0 i.e., K i = M p i for some i and have K i approaching either I no moves or K[−1 2, 12] periodic chain at fast enough rates. For instance, for i = 1, 2, . . . , take K 2i+1 = ‚ 1 − i −2 i −2 2i −2 1 − 2i −2 Œ , K 2i = ‚ i −2 1 − i −2 1 − 3i −2 3i −2 Œ . Then α 2i+1 = 45i 2 − 1 → ∞ and α 2i = −1 − 10i −2 92 − 10i −2 9 → −12 and there is no merging. 1486 Proposition 6.4. Let Q = {K[ α 1 , p 1 ], . . . , K[α m , p m ]} be a finite family of irreducible Markov kernels on V = {0, 1}. This family is merging in total variation if and only if K[−1 2, 12] 6∈ Q, that is α i , p i 6= −12, 12, i = 1, . . . , m. Proof. Recall that, by definition, Q is merging if any sequence made of kernels from Q is merging. If K[−1 2, 12] 6∈ Q then for any infinite sequence from Q either condition 2 or condition 3 of Proposition 6.2 is violated and we have merging. Remark 6.5. The family Q considered in Proposition 6.4 may not be merging in relative-sup distance even when K[−1 2, 12] 6∈ Q. Indeed, Remark 6.9 gives an example of kernels K[ α i , p i ], K[α j , p j ] 6= K[−12, 12] such that the product K[α i , p i ]K[α j , p j ] has an absorbing state at 0 and 1 is not absorbing. For any measure µ 0, the sequence K n ∞ 1 , with K n = K[α i , p i ] if n is odd and K n = K[α j , p j ] otherwise, satisfies lim n→∞ µ 2n 1 = 0. Remark 6.6. Consider a sequence K i ∞ 1 of Markov kernels on the two-point space such that 1 is satisfied. Write K i = K[α i , p i ]. Because the kernels are reversible, 1 and formula 2 yield |α i |1 + α i ≤ β 1. This implies that the α i ’s stay uniformly away from −1 2 and +∞. By Proposition 6.2 and 6 we get kK 0,n 0, · − K 0,n 1, ·k TV ≤ β n .

6.3 Stability