PROOF OF CORN 175
the performance of expert
(ω,ρ)
on the stationary and ergodic market sequence
X
0
, X
−1
, X
−2
,....
(ii) First, let the integers ω, ρ, and the vector s = s
−1
−ω
∈ R
dω
+
be fixed. From
Lemma B.5, we can get that the set {X
i
: 1 −j +ω ≤ i ≤ 0,
cov(X
i−1
i−ω
,s)
,
Va r (X
i−1
i−ω
)
√
Var(s)
≥ ρ}
can be expressed as {X
i
: 1 −j +ω ≤ i ≤ 0, E{(X
i−1
i−ω
−s)
2
}≤2Var(s)(1 −ρ).
Let P
(ω,ρ)
j,s
denote the (random) measure concentrated on {X
i
: 1 −j +ω ≤ i ≤
0, E{(X
i−1
i−ω
−s)
2
}≤2Var(s)(1 −ρ), defined by
P
(ω,ρ)
j,s
(A) =
i:1−j +ω≤i≤0,E{(X
i−1
i−ω
−s)
2
}≤2Var(s)(1−ρ)
II
A
(X
i
)
|{i : 1 −j +ω ≤ i ≤ 0, E{(X
i−1
i−ω
−s)
2
}≤2Var(s)(1 −ρ)}|
,A⊂ R
d
+
where II
A
denotes the indicator of function of the set A. If the above set of X
i
s is
empty, then let P
(ω,ρ)
j,s
= δ
(1,...,1)
be the probability measure concentrated on the vector
(1,...,1). In other words, P
(ω,ρ)
j,s
(A) is the relative frequency of the vectors among
X
1−j+ω
,...,X
0
that fall in the set A.
Observe that for all s, without probability 1,
P
(ω,ρ)
j,s
→ P
∗(ω,ρ)
s
=
P
X
0
|E{(X
−1
−ω
−s)
2
}≤2Var(s)(1−ρ)
if P(E{(X
−1
−ω
−s)
2
}≤2Var(s)(1 −ρ)) > 0
δ
(1,...,1)
if P(E{(X
−1
−ω
−s)
2
}≤2Var(s)(1 −ρ)) = 0
(B.2)
weakly as j →∞, where P
∗(ω,ρ)
s
denotes the limit distribution of P
(ω,ρ)
j,s
, and
P
X
0
|E{(X
−1
−ω
−s)
2
}≤2Var(s)(1−ρ)
denotes the distribution of the vector X
0
conditioned on
the event E{(X
−1
−ω
−s)
2
}≤2Var(s)(1 −ρ). To see this, let f be a bounded continuous
function defined on R
d
+
. Then, the ergodic theorem implies that if P(E{(X
−1
−ω
−s)
2
}≤
2Var(s)(1 −ρ)) > 0, then
f(x)P
∗(ω,ρ)
j,s
(dx) =
1
|1−j+ω|
i:1−j +ω≤i≤0,E{(X
i−1
i−ω
−s)
2
}≤2Var(s)(1−ρ)
f(X
i
)
1
|1−j+ω|
|{i:1−j +ω≤i≤0,E{(X
i−1
i−ω
−s)
2
}≤2Var(s)(1−ρ)}|
→
E{f(X
0
)II
{E{(X
−1
−ω
−s)
2
}≤2Var(s)(1−ρ)}
}
P{E{(X
−1
−ω
−s)
2
}≤2Var(s)(1−ρ)}
= E{f(X
0
)|E{(X
−1
−ω
−s)
2
}≤2Var(s)(1 −ρ)}
=
f(x)P
X
0
|E{(X
−1
−ω
−s)
2
}≤2Var(s)(1−ρ)
almost surely, as j →∞.
On the other hand, if P(E{(X
−1
−ω
−s)
2
}≤2Var(s)(1 −ρ)) = 0, then with proba-
bility 1, P
(ω,ρ)
j,s
is concentrated on (1,...,1) for all j, and
f(x)P
(ω,ρ)
j,s
(dx) =
f(1,...,1).
T&F Cat #K23731 — K23731_A002 — page 175 — 9/28/2015 — 20:47