Download A Strong Law of Large Numbers for Capacities

The Annals of Probability
2005, Vol. 33, No. 3, 1171–1178
DOI 10.1214/009117904000001062
© Institute of Mathematical Statistics, 2005
A STRONG LAW OF LARGE NUMBERS FOR CAPACITIES1
B Y FABIO M ACCHERONI AND M ASSIMO M ARINACCI
Università Bocconi and Università di Torino
We consider a totally monotone capacity on a Polish space and a
sequence of bounded p.i.i.d. random variables. We show that, on a full
set, any cluster point of empirical averages lies between the lower and the
upper Choquet integrals of the random variables, provided either the random
variables or the capacity are continuous.
1. Introduction. In this paper we prove a strong law of large numbers for
totally monotone capacities. Specifically, given a totally monotone capacity ν
defined on the Borel σ -algebra B of a Polish space , and a sequence {Xn }n≥1
of bounded, pairwise independent and identically distributed random variables, we
show that
ν
ω ∈ :
n
j =1 Xj (ω)
X1 dν ≤ lim inf
n
n
n
j =1 Xj (ω)
≤ lim sup
n
n
≤−
−X1 dν
= 1,
provided the Xn s are continuous or simple, or ν is continuous. In this way we
extend earlier results of Marinacci [13].
Under different names, totally monotone capacities have been widely studied
in both pure and applied mathematics. They have been introduced by Choquet [4]
motivated by some problems in potential theory, and in his wake many works have
studied them in both potential theory and probability theory (see, e.g., [6] and [8]).
In mathematical statistics and in mathematical economics, totally monotone capacities have been used to represent subjective prior beliefs when the information
on which such beliefs are based is not good enough to represent them by a standard
additive probability (see, e.g., [7, 11, 15, 20] and [19]).
Our result shows that even in a nonadditive setting, the limit behavior of
empirical averages has some noteworthy properties. In particular, we show that
eventually empirical averages lie, with probability one, between the lower and
upper Choquet integrals associated with the given capacity. This extends to the
Received February 2004; revised June 2004.
1 Supported by the Ministero dell’Istruzione, dell’Università e della Ricerca.
AMS 2000 subject classifications. 28A12, 60F15.
Key words and phrases. Capacities, Choquet integral, strong law of large numbers, contents,
measures, outer measures, strong theorems.
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F. MACCHERONI AND M. MARINACCI
nonadditive setting the classic Kolmogorov limit law, to which our result reduces
when ν is additive since in this case lower and upper Choquet integrals coincide.
In a subjective probability perspective, our result says that, while in the additive
case a Bayesian decision maker believes that the empirical averages of a sequence
of p.i.i.d. random variables tend to a given number, here he just believes that the
limit behavior of the empirical averages is confined in a given interval. This reflects
a possible lack of confidence in his probability assessments. This interpretation
of nonadditive limit laws and its relevance in mathematical economics has been
recently discussed at length in [9], to which we refer the interested reader for
details and references.
2. Preliminaries. Let be a Polish space and B its Borel σ -algebra.
A random variable (r.v.) is a (Borel) measurable function X : → R. A totally
monotone capacity on B is a set function ν : B → [0, 1] such that:
(c.1) ν(∅) = 0 and ν() = 1,
(c.2) ν(A) ≤ ν(B) for all Borel sets A ⊆ B,
(c.3) ν(Bn ) ↓ ν(B) for all sequences of Borel sets Bn ↓ B,
(c.4) ν(G
) ↑ ν(G) for
all sequences of open sets
G ↑ G,
n
n
(c.5) ν( nj=1 Bj ) ≥ ∅=J ⊆{1,...,n} (−1)|J |+1 ν( j ∈J Bj ) for every collection
B1 , . . . , Bn of Borel sets.
A set function ν : B → [0, 1] such that:
(c.6) ν(Bn ) ↑ ν() for all sequences of Borel sets Bn ↑ ,
is called continuous. A continuous set function ν : B → [0, 1] is a totally monotone
capacity if and only if (c.1), (c.2) and (c.5) hold (see [18] and [14], Theorem 10).
Let ν be a totally monotone capacity on B. As in the additive case, we say that
the elements of a sequence {Xn }n≥1 of r.v.s are pairwise independent with respect
to ν if, for each n, m ≥ 1 and for all open subsets Gn , Gm of R,
ν({Xn ∈ Gn , Xm ∈ Gm }) = ν({Xn ∈ Gn })ν({Xm ∈ Gm });
we say that they are identically distributed if, for each n, m ≥ 1 and each open
subset G of R,
ν(Xn ∈ G) = ν(Xm ∈ G).
The Choquet integral of a bounded r.v. X with respect to a totally monotone
capacity ν is defined by
X dν ≡
+∞
0
ν({X > t}) dt +
0
−∞
[ν({X > t}) − 1] dt.
The integrals on the right-hand side are Riemann integrals and they are well
defined since ν({X > t}) is a monotone function in t. The Choquet
integral is
positively homogeneous, monotone and translation invariant [i.e., (X + c) dν =
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SLLN FOR CAPACITIES
Xdν + c if c is constant]. It reduces to the standard integral when ν is an additive
probability measure.
holds for all r.v.s if and only if ν is
In general, X dν ≤ − −X dν. Equality
additive. The integrals X dν and − −X dν are sometimes called lower and
upper Choquet integrals, respectively.
3. The law of large numbers. We can now state our main result.
T HEOREM 1. Let ν be a totally monotone capacity on B, and {Xn }n≥1 a
sequence of bounded, pairwise independent and identically distributed random
variables. Then
n
j =1 Xj (ω)
ν ω ∈ : X1 dν ≤ lim inf
n
n
n
≤ lim sup
j =1 Xj (ω)
≤ − −X1 dν
n
provided at least one the following two conditions holds:
n
= 1,
(i) ν is continuous;
(ii) the random variables Xn are either continuous or simple.
A few remarks are in order. First, as ν(B) = 1 and A ⊆ B c imply ν(A) = 0,
under the assumptions of Theorem 1 we also have
n
ω ∈ : lim inf
ν
and
j =1 Xj (ω)
n
n
<
n
X1 dν
j =1 Xj (ω)
=0
> − −X1 dν = 0.
n
In other words,
with zero
probability empirical averages will eventually lie outside
the interval [ X1 dν, − −X1 dν].
Second, when ν is additive we have X1 dν = − −X1 dν, and so in this case
our result reduces to a standard Kolmogorov limit law
ν
ω ∈ : lim sup
n
n
j =1 Xj (ω)
= X1 dν = 1.
n
On the other hand, when ν is not additive in some cases it may happen (see [13])
that
n
n
j =1 Xj (ω)
j =1 Xj (ω)
< lim sup
= 1.
ν ω ∈ : lim inf
n
n
n
n
Finally, as anticipated, the closest existing theorem is due to [13]. Our result
is more general since [13] assumes that is compact, ν is continuous, the
ν
ω ∈ : lim
n
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F. MACCHERONI AND M. MARINACCI
r.v.s Xn are continuous, independent and that they satisfy some further technical
conditions. Moreover, the proof we provide is different and much simpler. In fact,
here we develop a technique that relies on the relations between totally monotone
capacities and correspondences, thus making it possible to use existing laws of
large numbers for correspondences. This approach might be useful in establishing
further generalizations of limit laws to the framework of capacities. This will be
the object of future research, along with the possibility of weakening some of the
continuity conditions assumed in Theorem 1.
4. Proof and related material. Denote by K (resp. G ) the class of all
nonempty compact subsets (resp. open subsets) of ; for the sake of completeness,
write B instead of B. If d is a Polish metric on , then K is a Polish space
when endowed with the Hausdorff metric
dH (K, L) ≡ max max min d(k, l), max min d(l, k) .
k∈K l∈L
l∈L k∈K
The Borel σ -algebra on K is also generated by the class {K ∈ K : K ⊆
G}G∈G .
4.1. Measurable correspondences and totally monotone capacities. Let
(I, C, λ) be a nonatomic and complete probability space. A (compact valued) correspondence F : I ⇒ is a map with domain I and whose values are nonempty
compact subsets of . For any A ⊆ , we put
F−1 (A) ≡ {s ∈ I : F (s) ⊆ A}.
A correspondence F : I ⇒ is measurable if F−1 (G) ∈ C for every G ∈ G . As
well known (see, e.g., [12]), the following facts are equivalent:
• F is measurable;
• F−1 (B) ∈ C for every B ∈ B ;
• F is measurable as a function F : I → K .
When a measurable correspondence F is regarded as a measurable function
F : I → K , we denote by F −1 its standard inverse image, that is,
F −1 (E ) ≡ {s ∈ I : F (s) ∈ E }
and by σ (F ) the σ -algebra generated by F , that is,
∀ E ⊆ K ,
σ (F ) ≡ F −1 (D) : D ∈ BK .
In the sequel we will need the next lemma, whose standard proof is omitted.
L EMMA 2.
Let F : I ⇒ be a measurable correspondence. Then
σ (F ) = σ {F−1 (G) : G ∈ G }
and {F−1 (G) : G ∈ G } is a π -class containing I .
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SLLN FOR CAPACITIES
A measurable function f : I → induces a probability distribution Pf on B
defined by
Pf (B) ≡ λ f −1 (B)
∀ B ∈ B .
In a similar way, a measurable correspondence F : I ⇒ induces a lower
distribution νF on B defined by
νF (B) ≡ λ F−1 (B)
∀ B ∈ B .
The next result, which links totally monotone capacities and lower distributions, is
essentially due to Choquet [4] (see also [15, 16] and [3]).
L EMMA 3. A set function ν : B → [0, 1] is a totally monotone capacity if
and only if there exists a measurable correspondence F : I ⇒ such that ν = νF .
A measurable selection of a correspondence F : I ⇒ is a measurable
function f : I → such that f (s) ∈ F (s) for almost all s ∈ I . The set of all
measurable selections of F is denoted by Sel F . The Aumann integral (see [2])
of a correspondence F : I ⇒ R with respect to λ is defined by
F dλ ≡
f dλ : f ∈ Sel F and f integrable .
If X : → R is continuous or simple, and F is a correspondence, then
(X ◦ F )(s) ≡ X(F (s)) is a correspondence [i.e., X(F (s)) ∈ KR for all s ∈ I ].
Moreover, since
(X ◦ F )−1 (A) = F−1 X −1 (A)
∀ A ⊆ R,
X ◦ F is measurable provided X and F are measurable.
L EMMA 4. Let F : I ⇒ be a measurable correspondence, and X : → R
be either bounded and continuous or simple and measurable. Then,
(X ◦ F ) dλ =
X dνF , −
−X dνF .
This is an immediate consequence of [3], Theorem 4.1.
4.2. Proof of Theorem 1. Suppose first that (ii) holds. By Lemma 3, there
exists a measurable correspondence F : I ⇒ such that ν = νF .
Next we show that the measurable correspondences {Xn ◦ F }n≥1 are pairwise
independent and identically distributed when regarded as measurable functions
Xn ◦ F : I → KR .
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F. MACCHERONI AND M. MARINACCI
Let n, m ≥ 1 and Gn , Gm ∈ GR , then
λ (Xn ◦ F )−1 (Gn ) ∩ (Xm ◦ F )−1 (Gm )
−1
= λ F−1 Xn−1 (Gn ) ∩ F−1 Xm
(Gm )
−1
= λ F−1 Xn−1 (Gn ) ∩ Xm
(Gm )
−1
= ν Xn−1 (Gn ) ∩ Xm
(Gm )
−1
= ν Xn−1 (Gn ) ν Xm
(Gm )
−1
= λ F−1 Xn−1 (Gn ) λ F−1 Xm
(Gm )
= λ (Xn ◦ F )−1 (Gn ) λ (Xm ◦ F )−1 (Gm ) .
This proves pairwise independence, since for all j = n, m, {(Xj ◦ F )−1 (G)}G∈GR
is a π -class containing I and generating the σ -algebra σ (Xj ◦ F ) (see Lemma 2).
Moreover, for each n, m ≥ 1, and each open subset G ∈ GR ,
λ (Xn ◦ F )−1 ({K ∈ KR : K ⊆ G})
= λ Xn ◦ F ∈ {K ∈ KR : K ⊆ G}
= λ (Xn ◦ F )−1 (G)
= λ F−1 Xn−1 (G)
= ν Xn−1 (G)
−1
= ν Xm
(G)
= λ (Xm ◦ F )−1 ({K ∈ KR : K ⊆ G}) .
This proves identical distribution since {K ∈ KR : K ⊆ G}G∈GR is a π -class
containing KR and generating BKR .
Clearly, for each n ≥ 1 and each h ∈ Sel Xn ◦ F , h dλ is finite (h is bounded);
moreover, by Lemma 4, Xn ◦ F dλ ∈ KR .
independent and identically distributed meaIn sum, {Xn ◦ F }n≥1 are pairwise
surable correspondences with Xn ◦ F dλ ∈ KR for all n ≥ 1. A generalization
due to [10] (see also [5] and [17]) of a result of Artstein and Vitale [1] guarantees
that
λ
n
1
s∈I:
Xj (F (s)) →
n j =1
X1 ◦ F dλ
= 1.
By Lemma 4,
X1 ◦ F dλ=
X1 dν, −
−X1 dν .
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SLLN FOR CAPACITIES
Let an (ω) =
n
j =1 Xj (ω)
n
and set
n
1
Xj (F (s)) →
S1 ≡ s ∈ I :
n j =1
S2 ≡ s ∈ I :
X1 dν, −
2 ≡ ω ∈ :
−X1 dν
,
X1 dν ≤ lim inf an (ω)
n
≤ lim sup an (ω) ≤ −
−X1 dν ∀ ω ∈ F (s) ,
n
X1 dν ≤ lim inf an (ω) ≤ lim sup an (ω) ≤ −
n
n
We want to show that ν(2 ) = 1. Notice that
−X1 dν .
ν(2 ) = λ {s ∈ I : F (s) ⊆ 2 } = λ(S2 ).
The next claim will be used to show that S1 ⊆ S2 .
C LAIM 1.
Let {Kn } be a sequence in KR such that Kn → [α, β]. Then,
α ≤ lim inf kn ≤ lim sup kn ≤ β
n
n
for each sequence {kn } in R such that kn ∈ Kn for all n ≥ 1.
P ROOF. By definition of Hausdorff metric, Kn converges to [α, β] if and
only if max(maxtn ∈Kn minr∈[α,β] |tn − r|, maxr∈[α,β] mintn ∈Kn |r − tn |) → 0, in
particular
max min |tn − r| → 0.
(1)
tn ∈Kn r∈[α,β]
/ [α, β],
Let {knj } be a subsequence of {kn } such that knj → ∈ [−∞, +∞]. If ∈
then there exists ε > 0 such that eventually |knj − r| > ε for all r ∈ [α, β]. Hence,
we have eventually minr∈[α,β] |knj − r| > ε, thus contradicting (1). If s ∈ S1 , then n1 nj=1 Xj (F (s)) → [ X1 dν, − −X1 dν]. Hence, for all ω ∈
F (s), we have an (ω) = n1 nj=1 Xj (ω) ∈ n1 nj=1 Xj (F (s)); by Claim 1,
X1 dν ≤ lim inf an (ω) ≤ lim sup an (ω) ≤ −
n
n
−X1 dν.
Therefore, S1 ⊆ S2 and so ν(2 ) = λ(S2 ) ≥ λ(S1 ) = 1. This completes the proof
of the result when (ii) holds.
As to (i), denoting by τ the Polish topology on , there exists a Polish topology
τ ∗ ⊇ τ on such that σ (τ ∗ ) = B , and such that all the Xn s are τ ∗ -continuous
(see, e.g., [21]). Since ν is continuous, then it is a totally monotone capacity with
respect to the topology τ ∗ ; we can thus assume that (ii) holds.
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F. MACCHERONI AND M. MARINACCI
Acknowledgments. We thank Steve Lalley (the Editor) and an anonymous
referee for helpful suggestions.
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