Optimality

Optimal sampling strategies 285
Since δi > 0∀i, for the concavity of δi it suffices to show that
�
�
1−βzi
(7.22)
κi+2 − κi+1 ≤ 0.
1−βzi+2
Now
(7.23)
1−αzi
1−αzi+2
− 1−βzi
1−βzi+2
= (α−β)(zi+2− zi)
(1−αzi+2)(1−βzi+2)
≥ 0.
Then (7.20) and (7.23) combined with the fact that κi≥ 0,∀i proves (7.22). �
Proof of Theorem 3.1. We split the theorem into three claims.
Claim (1): L ∗ :=∪kL (k) (x ∗ k )∈Lγ(n).
From (3.10), (3.11), and (3.13) we obtain
(7.24)
µγ(n) + Pγ− 1
var(Vγ)
E(Vγ|Lγk)
L∈Λγ(n)
k=1
−1
= max
Pγ �
≤ max
Pγ �
X∈∆n(Nγ1,...,NγPγ )
µγ,γk(xk).
k=1
Clearly L ∗ ∈ Λγ(n). We then have from (3.10) and (3.11) that
(7.25)
µγ(n) + Pγ− 1
var(Vγ) ≥ E(Vγ|L ∗ ) −1 + Pγ− 1
var(Vγ) =
=
Pγ �
k=1
Thus from (7.25) and (7.26) we have
(7.26) µγ(n) =E(Vγ|L ∗ ) −1 = max
which proves Claim (1).
Pγ �
k=1
µγ,γk(x ∗ k) = max
X∈∆n(Nγ1,...,NγPγ )
k=1
E(Vγ|L ∗ γk) −1
Pγ �
X∈∆n(Nγ1,...,NγPγ )
µγ,γk(xk).
k=1
Pγ �
µγ,γk(xk)− Pγ− 1
var(Vγ) ,
Claim (2): If L∈Lγk(n) then L∈Lγ,γk(n) and vice versa.
Denote an arbitrary leaf node of the tree of γk as E. Then Vγ, Vγk, and E are
related through
(7.27) Vγk = ϱγkVγ + Wγk,
and
(7.28) E = ηVγk + F
where η and ϱγk are scalars and Wγk, F and Vγ are independent random variables.
We note that by definition var(Vγ) > 0∀γ (see Definition 2.5). From Lemma 7.1