Non-monotone convergence in the quadratic Wasserstein distance
NON-MONOTONE CONVERGENCE IN THE QUADRATIC
WASSERSTEIN DISTANCE
WALTER SCHACHERMAYER, UWE SCHMOCK, AND JOSEF TEICHMANN
Abstract. We give an easy counter-example to Problem 7.20 from C. Vil-
lani’s book on mass transport: in general, the quadratic Wasserstein distance
between n-fold normalized convolutions of two given measures fails to decrease
monotonically.
We use the terminology and notation from [5]. For Borel measures µ, ν on Rd
we define the quadratic Wasserstein distance
T (µ, ν) := inf
(X,Y )
‖X − Y ‖2
where ‖ · ‖ is the Euclidean distance on Rd and the pairs (X,Y ) run through all
random vectors defined on some common probabilistic space (Ω,F ,P), such that X
has distribution µ and Y has distribution ν. By a slight abuse of notation we define
T (U, V ) := T (µ, ν) for two random vectors U , V , such that U has distribution µ
and V has distribution ν. The following theorem (see [5, Proposition 7.17]) is due
to Tanaka [4].
Theorem 1. For a, b ∈ R and square integrable random vectors X, Y , X ′, Y ′
such that X is independent of Y , and X ′ is independent of Y ′, and E[X ] = E[X ′]
or E[Y ] = E[Y ′], we have
T (aX + bY, aX ′ + bY ′) ≤ a2T (X,X ′) + b2T (Y, Y ′).
For a sequence of i.i.d. random vectors (Xi)i∈N we define the normalized partial
Sm :=
Xi, m ∈ N.
If µ denotes the law of X1, we write µ
(m) for the law of Sm. Clearly µ
(m) equals,
up to the scaling factor
m, the m-fold convolution µ ∗ µ ∗ · · · ∗ µ of µ.
We shall always deal with measures µ, ν with vanishing barycenter. Given two
measures µ and ν on Rd with finite second moments, we let (Xi)i∈N and (X
i)i∈N
be i.i.d. sequences with law µ and ν, respectively, and denote by Sm and S
m the
corresponding normalized partial sums. From Theorem 1 we obtain
µ(2m), ν(2m)
µ(m), ν(m)
, m ∈ N,
Date: October 4, 2006.
Financial support from the Austrian Science Fund (FWF) under grant P 15889, from the
Vienna Science and Technology Fund (WWTF) under grant MA13, from the European Union
under grant HPRN-CT-2002-00281 is gratefully acknowledged. Furthermore this work was finan-
cially supported by the Christian Doppler Research Association (CDG) via PRisMa Lab. The
authors gratefully acknowledge a fruitful collaboration and continued support by Bank Austria
Creditanstalt (BA-CA) and the Austrian Federal Financing Agency (ÖBFA) through CDG.
http://arxiv.org/abs/0704.0876v1
http://www.fwf.ac.at/
http://www.wwtf.at/
http://www.cdg.ac.at/
http://www.prismalab.at/
http://www.ba-ca.com/
http://www.oebfa.co.at/
2 WALTER SCHACHERMAYER, UWE SCHMOCK, AND JOSEF TEICHMANN
from which one may quickly deduce a proof of the Central Limit Theorem (compare
[5, Ch. 7.4] and the references given there).
However, we can not deduce from Theorem 1 that the inequality
(1) T
µ(m+1), ν(m+1)
µ(m), ν(m)
holds true for all m ∈ N. Specializing to the case m = 2, an estimate, which we
can obtain from Tanaka’s Theorem, is
µ(3), ν(3)
µ(2), ν(2)
+ T (µ, ν)
≤ T (µ, ν).
This contains some valid information, but does not imply (1). It was posed as
Problem 7.20 of [5], whether inequality (1) holds true for all probability measures
µ, ν on Rd and all m ∈ N.
The subsequent easy example shows that the answer is no, even for d = 1 and
symmetric measures. We can choose µ = µn and ν = νn for sufficiently large n ≥ 2,
as the proposition (see also Remark 1) shows.
Proposition 1. Denote by µn the distribution of
∑2n−1
i=1 Zi, and by νn the distri-
bution of
i=1 Zi with (Zi)i∈N i.i.d. and P(Z1 = 1) = P(Z1 = −1) =
. Then
(2) lim
n T (µn ∗ µn, νn ∗ νn) =
while T (µn ∗ µn ∗ µn, νn ∗ νn ∗ νn) ≥ 1 for all n ∈ N.
Remark 1. If one only wants to find a counter-example to Problem 7.20 of [5],
one does not really need the full strength of Proposition 1, i.e. the estimate that
T (µn ∗ µn, νn ∗ νn) = O(1/
n). In fact, it is sufficient to consider the case n = 2
in order to contradict the monotonicity of inequality (1). Indeed, a direct calculation
reveals that
T (µ2 ∗ µ2, ν2 ∗ ν2) = 0.625 <
T (µ2 ∗ µ2 ∗ µ2, ν2 ∗ ν2 ∗ ν2).
Proof of Proposition 1. We start with the final assertion, which is easy to show.
The 3-fold convolutions of the measures µn and νn, respectively, are supported
on odd and even numbers, respectively. Hence they have disjoint supports with
distance 1 and so the quadratic transportation costs are bounded from below by 1.
For the proof of (2), fix n ∈ N, define σn = µn ∗ µn and τn = νn ∗ νn, and note
that σn and τn are supported by the even numbers. For k = −(2n−1), . . . , (2n−1)
we denote by pn,k the probability of the point 2k under σn, i.e.
pn,k =
4n− 2
k + 2n− 1
24n−2
We define pn,k = 0 for |k| ≥ 2n. We have τn = σn ∗ ρ, where ρ is the distribution
giving probability 1
to −2, 0, 2, respectively. We deduce that for 0 ≤ k ≤
2n− 2,
τn(2k + 2) =
pn,k +
pn,k+2 +
pn,k+1
(pn,k − pn,k+1) +
(pn,k+2 − pn,k+1) + σn(2k + 2)
1− pn,k+1
pn,k+1
(pn,k+2
pn,k+1
+ σn(2k + 2).
NON-MONOTONE CONVERGENCE IN THE QUADRATIC WASSERSTEIN DISTANCE 3
Notice that pn,k ≥ pn,k+1 for 0 ≤ k ≤ 2n− 1. The term in the first parentheses is
therefore non-negative. It can easily be calculated and estimated via
0 ≤ 1− pn,k+1
k+2n−1
) = 1− 2n− k − 1
k + 2n
2k + 1
2n+ k
≤ 2k + 1
for 0 ≤ k ≤ 2n− 1.
Following [5] we know that the quadratic Wasserstein distance T can be given
by a cyclically monotone transport plan π = πn. We define the transport plan π
via an intuitive transport map T . It is sufficient to define T for 0 ≤ k ≤ 2n − 1,
since it acts symmetrically on the negative side. T moves mass 1
from the
point 2k to 2k+ 2 for k ≥ 1. At k = 0 the transport T moves 1
pn,0 to every side,
which is possible, since there is enough mass concentrated at 0.
By equation (3) we see that the transport T moves σn to τn, since, for 1 ≤
k ≤ 2n − 2, the first terms corresponds to the mass, which arrives from the left
and is added to σn, and the second term to the mass, which is transported away:
summing up one obtains τn. For k = 2n − 1, mass only arrives from the left. At
k = 0 mass is only transported away. By the symmetry of the problem around 0
and by the quadratic nature of the cost function (the distance of the transport is 2,
hence cost 22), we finally have
T (σn, τn) ≤ 2
2k + 1
2n+ k
2k + 1
By the Central Limit Theorem and uniform integrability of the function x 7→ x+ :=
max(0, x) with respect to the binomial approximations, we obtain
(2k)pn,k =
2/2 dx.
Hence
lim sup
n T (σn, τn) ≤
≈ 0.79788.
In order to obtain equality we start from the local monotonicity of the respective
transport maps on non-positive and non-negative numbers. It easily follows that
the given transport plan is cyclically monotone and hence optimal (see [5, Ch. 2]).
The subsequent equality allows also to consider estimates from below. Rewriting
(3) yields
τn(2k + 2) =
pn,k+1
( pn,k
pn,k+1
pn,k+2
1− pn,k+1
pn,k+2
+ σn(2k + 2)
for 0 ≤ k ≤ 2n− 3, and
τn(2k + 2) =
pn,k+1
( pn,k
pn,k+1
+ σn(2k + 2)
for k = 2n− 2. Furthermore,
pn,k+1
− 1 =
k+2n−1
) − 1 = k + 2n
2n− k − 1
− 1 = 2k + 1
2n− k − 1
≥ 2k + 1
4 WALTER SCHACHERMAYER, UWE SCHMOCK, AND JOSEF TEICHMANN
for 0 ≤ k ≤ 2n− 2. This yields by a reasoning similar to the above that
T (σn, τn) ≥
pn,k+1
2k + 1
hence
lim inf
n T (σn, τn) ≥
Remark 2. Let p ≥ 2 be an integer. By slight modifications of the proof of Propo-
sition 1 we can construct sequences of measures (µn)n∈N and (νn)n∈N, such that
the quadratic Wasserstein distances of k-fold convolutions are bounded from below
by 1 for all k which are not multiples of p, while
T (µ(p)n , ν(p)n ) = 0.
Remark 3. Assume the notations of [5]. In the previous considerations we can
replace the quadratic cost function by any other lower semi-continuous cost function
c : R2 → [0,+∞], which is bounded on parallels to the diagonal and vanishes on
the diagonal. For example, if we choose c(x, y) = |x− y|r for 0 < r < ∞, then we
obtain the same asymptotics as in Proposition 1 (with a different constant).
Remark 4. We have used in the above proof that τn is obtained from σn by con-
volving with the measure ρ. In fact, this theme goes back (at least) as far as L.
Bachelier’s famous thesis from 1900 on option pricing [2, p. 45]. Strictly speaking,
L. Bachelier deals with the measure assigning mass 1
to −1, 1 and considers con-
secutive convolutions, instead of the above ρ. Hence convolutions with ρ correspond
to Bachelier’s result after two time steps. Bachelier makes the crucial observation
that this convolution leads to a radiation of probabilities: Each stock price x radi-
ates during a time unit to its neighboring price a quantity of probability proportional
to the difference of their probabilities. This was essentially the argument which al-
lowed us to prove (1). Let us mention that Bachelier uses this argument to derive
the fundamental relation between Brownian motion (which he was the first to define
and analyse in his thesis) and the heat equation (compare e.g. [3] for more on this
topic).
Remark 5. Having established the above counterexample, it becomes clear how
to modify Problem 7.20 from [5] to give it a chance to hold true. This possible
modification was also pointed out to us by C. Villani.
Problem 1. Let µ be a probability measure on Rd with finite second moment
and vanishing barycenter, and γ the Gaussian measure with same first and second
moments. Does (T (µ(n), γ))n≥1 decrease monotonically to zero?
When entropy is considered instead of the quadratic Wasserstein distance the
corresponding question on monotonicity was answered affirmatively in the recent
paper [1].
One may also formulate a variant of Problem 7.20 as given in (1) by replacing
the measure ν through a log-concave probability distribution. This would again
generalize problem 1.
NON-MONOTONE CONVERGENCE IN THE QUADRATIC WASSERSTEIN DISTANCE 5
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strasse 8–10, A-1040 Vienna, Austria.
http://www.numdam.org/en/
http://www.fam.tuwien.ac.at/
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