Appendix B
Proofs and Derivations
B.1 Proof of CORN
B.1.1 Proof of Theorem 1
∗
In this appendix, we give a detailed proof that the portfolio scheme correlation-driven
nonparametric learning (CORN) (Li et al. 2011a) is universal with respect to the
class of all ergodic processes. We first give a concise definition about “universal”
considered in this note.
Definition 1 An investment strategy B is called universal with respect to a class of
stationary and ergodic processes {X
n
}
+∞
−∞
, if, for each process in the class,
lim
n→∞
1
n
log S
n
(B) = W
∗
almost surely.
Before we give the theorem and its proof, we introduce some necessary lemmas.
Lemma B.1 (Breiman 1957 [Correction version 1960]) Let Z ={Z
i
}
∞
−∞
be a sta-
tionary and ergodic process. For each positive integer i, let T
i
denote the operator
that shifts any sequence {...,z
−1
,z
0
,z
1
,...}by i digits to the left. Let f
1
,f
2
,...be
a sequence of real-valued functions such that lim
n→∞
f
n
(Z) = f(Z)almost surely
for some function f . Assume that E sup
n
|f
n
(Z)| < ∞. Then,
lim
n→∞
1
n
n
i=1
f
i
(T
i
Z) = Ef(Z) almost surely.
Lemma B.2 (Algoet and Cover 1988) Let Q
n∈N∪{∞}
be a family of regular proba-
bility distributions over the set R
d
+
of all market vectors such that E{|log U
(j)
n
|} < ∞
for any coordinate of a random market vector U
n
= (U
(1)
n
,...,U
(d)
n
) distributed
according to Q
n
. In addition, let B
∗
(Q
n
) be the set of all log-optimal portfolios with
∗
The proof idea is mainly provided by Vladimir Vovk, and the proof is then finished by Dingjiang
Huang and Bin Li.
171
T&F Cat #K23731 — K23731_A002 — page 171 — 9/28/2015 — 20:47
172 PROOFS AND DERIVATIONS
respect to Q
n
, that is, the set of all portfolios b that attain max
b∈
d
E{logb, U
n
}.
Consider an arbitrary sequence b
n
∈ B
∗
(Q
n
).If
Q
n
→ Q
∞
weakly as n →∞,
then, for Q
∞
-almost all u,
lim
n→∞
b
n
, u→b
∗
, u,
where the right-hand side is constant as b
∗
ranges over B
∗
(Q
∞
).
Lemma B.3 (Algoet and Cover 1988) Let X be a random market vector defined on
a probability space(, F, P) satisfying E{|log X
(j)
|} < ∞.IfF
k
is an increasing
sequence of sub-σ-fields of F with:
F
k
F
∞
⊆ F,
then
E
max
b
E[logb, X|F
k
]
E
max
b
E[logb, X|F
∞
]
,
as k →∞ where the maximum on the left-hand side is taken on over all
F
k
-measurable functions b, and the maximum on the right-hand side is taken on
over all F
∞
-measurable functions b.
Lemma B.4 Let μ be the Lebesgue measure on the Euclidean space R
n
and A
be a Lebesgue measurable subset of R
n
. Define the approximate density of A in
a ε-neighborhood of a point x in R
n
as
d
ε
(x) =
μ(A ∩B
ε
(x))
μ(B
ε
(x))
,
where B
ε
denotes the closed ball of radius ε centered at x. Then for almost every point
x of A the density,
d(x) = lim
ε→0
d
ε
(x)
exists and is equal to 1.
Lemma B.5 The inequality
cov(X, X
)
√
Var (X)
√
Var (X
)
≥ ρ,
which describes the similarity of X and X
in CORN strategy, is approximately
equivalent to
2Var(X)(1 −ρ) ≥ E{(X −X
)
2
}.
T&F Cat #K23731 — K23731_A002 — page 172 — 9/28/2015 — 20:47
PROOF OF CORN 173
Proof In general, from the covariance cov(X, X
) it is impossible to derive a topol-
ogy, since cov(X, X
) = 1 does not imply that E{(X −X
)
2
}=0. However, because X
and X
are relative prices, then we have E{(X −X
)
2
}≈0. For the Euclidean
distance, we have that
E{(X −X
)
2
}=Var (X −X
) +(E{X −X
})
2
= Va r (X) −2cov(X, X
) +Var (X
) +(E{X −X
})
2
.
Thus, the similarity means that
Var (X) +Var (X
) +(E{X −X
})
2
−E{(X −X
)
2
}
√
Var (X)
√
Var (X
)
≥ 2ρ
or, equivalently,
Var (X) +Var (X
) +(E{X −X
})
2
−2ρ
Var (X)
Var (X
) ≥ E{(X −X
)
2
}.
Since both Va r (X) and |E{X −X
}| have the same order of magnitude,
∗
they are in
the range 10
−4
, 10
−3
; therefore, the previous inequality approximately means that
2Var(X)(1 −ρ) ≥ E{(X −X
)
2
}.
Lemma B.6 Assume that x
1
, x
2
,... are the realizations of the random vectors
X
1
, X
2
,... drawn from the vector-valued stationary and ergodic process {X
n
}
∞
−∞
.
The fundamental limits (determined in Algoet 1992, 1994; Algoet and Cover 1988),
reveal that the so-called log-optimum portfolio B
∗
={b
∗
(·)} is the best possible
choice. More precisely, in trading period n, let b
∗
(·) be such that
E{logb
∗
(X
n−1
1
), X
n
|X
n−1
1
}=max
b(·)
E{logb(X
n−1
1
), X
n
|X
n−1
1
}.
If S
∗
n
= S
n
(B
∗
) denotes the capital achieved by a log-optimum portfolio strategy B
∗
,
after n trading periods, then for any other investment strategy B with capital S
n
=
S
n
(B) and for any stationary and ergodic process {X
n
}
∞
−∞
,
lim sup
n→∞
1
n
log
S
n
S
∗
n
≤ 0 almost surely
and
lim
n→∞
1
n
log S
∗
n
= W
∗
almost surely,
where
W
∗
= E
max
b(·)
E{logb(X
−1
−∞
), X
0
|X
−1
−∞
}
,
is the maximal possible growth rate of any investment strategy.
∗
See Appendix C for more details.
T&F Cat #K23731 — K23731_A002 — page 173 — 9/28/2015 — 20:47
174 PROOFS AND DERIVATIONS
Now, we give the universal theorem and its proof.
Theorem B.1 The portfolio scheme CORN is universal with respect to the class of
all ergodic processes such that E{|log X
(j)
|} < ∞, for j = 1, 2,...,d.
Proof To prove that the strategy CORN is universal with respect to the class of all
ergodic processes, we need to prove that if, for each process in the class,
lim
n→∞
1
n
log S
n
(B) = W
∗
almost surely,
where B denote the strategy CORN; and
W
∗
= lim
n→∞
1
n
log S
∗
n
= E{max
b(·)
E{logb(X
−1
−∞
), X
0
|X
−1
−∞
}}.
We divide the proof into three parts.
(i) According to Lemma B.6, we know that lim
n→∞
1
n
log S
n
−
1
n
log S
∗
n
≤ 0, then
lim
n→∞
1
n
log S
n
≤ lim
n→∞
1
n
log S
∗
n
= W
∗
. So it suffices to prove that
lim inf
n→∞
W
n
(B) = lim inf
n→∞
1
n
log S
n
(B) ≥ W
∗
almost surely.
Without loss of generality, we may assume S
0
= 1, so that
W
n
(B) =
1
n
log S
n
(B)
=
1
n
log
ω,ρ
q
ω,ρ
S
n
(
(ω,ρ)
)
≥
1
n
log
sup
ω,ρ
q
ω,ρ
S
n
(
(ω,ρ)
)
=
1
n
sup
ω,ρ
(log q
ω,ρ
+log S
n
(
(ω,ρ)
))
= sup
ω,ρ
W
n
(
(ω,ρ)
) +
log q
ω,ρ
n
.
Thus,
lim inf
n→∞
W
n
(B) = lim inf
n→∞
sup
ω,ρ
W
n
(
(ω,ρ)
) +
log q
ω,ρ
n
≥ sup
ω,ρ
lim inf
n→∞
W
n
(
(ω,ρ)
) +
log q
ω,ρ
n
= sup
ω,ρ
lim inf
n→∞
W
n
(
(ω,ρ)
). (B.1)
The simple argument above shows that the asymptotic rate of growth of the
strategy B is at least as large as the supremum of the rates of growth of all elemen-
tary strategies
(ω,ρ)
. Thus, to estimate lim inf
n→∞
W
n
(B), it suffices to investigate
T&F Cat #K23731 — K23731_A002 — page 174 — 9/28/2015 — 20:47
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
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