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
The study of stability on the 2-point space turns out to be quite interesting. We prove the following result which readily follows from the more precise statement in Proposition 6.10 below.
Theorem 6.7. Fix 0 ε ≤ η ≤ 12. Let Qε, η be the set of all Markov kernels K[α, p] on {0, 1}
with p ∈ [
η, 1 − η] and α ∈ [− minp, 1 − p + ε, ∞. Then Q
ε, η is ε
2
η
−1
-stable with respect to any measure µ
with µ
0 ∈ [η, 1 − η]. The special case in the following proposition is easy but important for the treatment of the general
case in Theorem 6.7.
Proposition 6.8. Fix η ∈ 0, 12 and let
Q+, η = {K[α, p] : α ≥ 0, p ∈ [η, 1 − η]}. Then Q+,
η is η
−1
-stable with respect to any measure µ
with µ
0 ∈ [η, 1 − η]. Proof. The crucial point is that when all
α
i
are non-negative then p
0,n
is a convex combination of the p
i
, 1 ≤ i ≤ n. Hence p
0,n
∈ [η, 1 − η] and µ
n
0 ∈ [η, 1 − η]. This gives the stated result.
1487
When the α
i
are not all positive it is still possible to show c-stability but the proof is bit subtle. This illustrates in this simple case the intrinsic difficulties related to the notion of stability.
Remark 6.9. Consider the product K[ α
1
, p
1
]K[α
2
, p
2
] = K[α
0,2
, p
0,2
] when p
1
6= p
2
, minp
1
, 1 − p
1
= 1 − p
1
, minp
2
, 1 − p
2
= p
2
, α
1
= −1 − p
1
and α
2
= −p
2
. Then p
0,2
= α
2
p
1
+ 1 + α
1
p
2
1 + α
1
+ α
2
= −p
2
p
1
+ p
1
p
2
p
2
− p
1
= 0.
Proposition 6.10. Let K
i
= K[α
i
, p
i
], i = 1, 2, . . . , be a sequence of Markov kernels on {0, 1} with K
i
∈ Qε, η. Then p
0,n
∈ [ε
2
η, 1 − ε
2
η] for all n ≥ 1. In order to prove this proposition, we need the following technical lemma.
Lemma 6.11. Fix 0 ≤ ε ≤ η ≤ 12. Let K
1
= K[α
1
, p
1
] and K
2
= K[α
2
, p
2
] be two Markov kernels on {0, 1}. Assume that p
i
∈ [η, 1 − η], i = 1, 2. 1 If
α
1
≥ 0 and α
2
≥ 0 then p
0,2
∈ [η, 1 − η]. 2 If
α
1
∈ [− minp
1
, 1 − p
1
+ ε, 0] and α
2
≥ 0 then α
0,2
∈ [− minp
0,2
, 1 − p
0,2
+ ε, 0] and p
0,2
∈ [η, 1 − η]. 3 If
α
1
∈ [− minp
1
, 1 − p
1
+ ε, ∞ and α
2
∈ [− minp
2
, 1 − p
2
+ ε, 0] then p
0,2
∈ [εη, 1 − εη]. Proof.
1 The fact that p
0,2
∈ [η, 1 − η] follows since p
0,2
is a convex combination of p
1
and p
2
and α
0,2
≥ 0 follows by Lemma 6.1.
2 Since 1 + α
1
≥ 0 then p
0,2
is a convex combination of p
1
and p
2
so we get p
0,2
∈ [η, 1 − η]. To see that
α
0,2
∈ [0, − minp
0,2
, 1 − p
0,2
+ ε], Lemma 6.1 implies that we just need to check the following two inequalities:
a | α
1
α
2
| ≤ p
1
α
2
+ 1 + α
1
p
2
− ε1 + α
1
+ α
2
. This inequality follows from p
1
α
2
+ 1 + α
1
p
2
− ε1 + α
1
+ α
2
+ α
1
α
2
= α
2
p
1
+ α
1
+ 1 + α
1
p
2
− ε1 + α
1
+ α
2
≥ εα
2
+ 1 + α
1
p
2
− ε1 + α
1
+ α
2
= 1 + α
1
p
2
− ε ≥ 1 + α
1
η − ε ≥ 0. b |
α
1
α
2
| ≤ q
1
α
2
+ 1 + α
1
q
2
− ε1 + α
1
+ α
2
where q
i
= 1 − p
i
. This inequality follows from the same calculations as in part a and the facts that q
i
∈ [η, 1 − η] and α
i
≥ −q
i
+ ε.
3 Write
p
0,2
= α
2
p
1
+ 1 + α
1
p
2
1 + α
1
+ α
2
= p
2
− |α
2
| p
1
− p
2
1 + α
1
+ α
2
7 and observe that
p
1
− p
2
1 + α
1
+ α
2
= |p
1
− p
2
| 1 +
α
1
+ α
2
≤ |p
1
− p
2
| |p
1
− p
2
| + 2ε 1488
The last inequality follows from 1 + α
1
+ α
2
≥ p
1
− p
2
+ 2ε and 1 + α
1
+ α
2
≥ p
2
− p
1
+ 2ε. Note that x
x + 2ε is an increasing function for x ≥ 0, since |p
1
− p
2
| ≤ 1 we get that p
1
− p
2
1 + α
1
+ α
2
≤ 1
1 + 2 ε
≤ 1 − ε. If p
1
≤ p
2
then 7 implies p
0,2
≥ p
2
≥ η ≥ εη. With q
2
= 1 − p
2
, we get p
0,2
≤ 1 − q
2
+ q
2
p
2
− p
1
1 + α
1
+ α
2
≤ 1 − q
2
+ q
2
1 − ε ≤ 1 − εq
2
≤ 1 − εη. If p
2
≤ p
1
then 7 implies p
0,2
≤ p
2
≤ 1 − η ≤ 1 − εη. For the lower bound we note that p
0,2
≥ p
2
− p
2
p
1
− p
2
1 + α
1
+ α
2
≥ p
2
− p
2
1 − ε ≥ εp
2
≥ εη.
Proof of Proposition 6.10. To introduce convenient notation, if K[ α, p] ∈ M
p
we say that α = αK
and p = pK. Let K
i ∞
1
be the sequence of kernels considered in Proposition 6.10. Fix n ≥ 1, and let {i
j
}
m j=1
be the set of numbers 1 ≤ i
j
≤ n such that αK
i
j
0. Set i = 0 and consider the kernels
Q
j
= K
i
j
,i
j+1
−1
for 0 ≤ j ≤ m Note that for any j ∈ [1, m] and l ∈ [i
j
, i
j+1
− 1] we have that αK
l
≥ 0 and pK
l
∈ [η, 1 − η]. Note that either Q
j
= I or by Lemma 6.11 we have that αQ
j
≥ 0 and pQ
j
∈ [η, 1 − η]. Now we write,
K
0,n
= Q K
i
1
Q
1
. . . K
i
j
Q
j
. . . K
i
m
Q
m
. For any j ∈ [1, m], consider the kernel M
j
= K
i
j
Q
j
. Since αK
j
∈ [− minp
i
, 1 − p
i
+ ε, 0] and αQ
j
≥ 0 or Q
j
= I, it follows by Lemma 6.11 that
αM
j
∈ [− minpM
j
, 1 − pM
j
+ ε, 0] and pM
j
∈ [η, 1 − η]. 8
So now we write K
0,n
= Q M
1
M
2
. . . M
m
. Consider the kernels e
M
i
= M
2i−1
M
2i
. It follows by Lemma 6.11 and 8 that α e
M
i
≥ 0 and p e M
j
∈ [εη, 1 − εη]. Let B = Q
Ý M
1
. Since Q = I or αQ
≥ 0 and pQ ∈ [η, 1 − η] by Lemma 6.11 we
have that αB ≥ 0 and pB ∈ [εη, 1 − εη]. If m = 2k then we write
K
0,n
= B e M
2
. . . e M
k
. For any K ∈ {B, e
M
2
, . . . , e M
k
} we have that αK ≥ 0 and pK ∈ [εη, 1 − εη], so by Lemma 6.11 we get
αK
0,n
≥ 0 and pK
0,n
∈ [εη, 1 − εη]. If m = 2k + 1 the we write
K
0,n
= B e M
2
. . . e M
k
M
m
= M
′
M
m
where M
′
= B e M
2
. . . e M
k
. By the same arguments as above we have αM
′
≥ 0 and pM
′
∈ [εη, 1 − εη]. Lemma 6.11 and 8 we get that
pK
0,n
∈ [ε
2
η, 1 − ε
2
η] as desired.
1489
6.4 Stability and relative-sup merging
