Second Zolotarev Distance Overview

• The second Zolotarev distance is a second-order probability metric defined using smooth test functions with bounded second derivatives to capture curvature constraints.
• It establishes sharp inequalities with the quadratic Wasserstein distance, offering precise bounds under variance constraints.
• Its formulations extend to both one-dimensional and multivariate settings, integrating optimal transport principles, martingale couplings, and quantitative limit theorems.

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to=arxiv_search.search 大发云json content={"query":"id:2511.00232 OR title:\"Sharp inequalities between Zolotarev and Wasserstein distances in P2(R^d)\"","max_results":5,"sort_by":"submittedDate"}大奖吗
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to=arxiv_search.search 北京赛车计划json content={"query":"id:2401.04879 OR title:\"From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment\"","max_results":5,"sort_by":"submittedDate"}
to=arxiv_search.search 天天中彩票出票json content={"query":"id:2210.04060 OR title:\"A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal\"","max_results":5,"sort_by":"submittedDate"}
to=arxiv_search.search 大发快三走势图json content={"query":"id:1911.11882 OR title:\"Bounding Zolotarev numbers using Faber rational functions\"","max_results":5,"sort_by":"submittedDate"}
The second Zolotarev distance is a second-order probability metric defined through smooth test functions with controlled curvature. In the one-dimensional literature it is usually written (\zeta_2), while in recent multivariate work it also appears as (Z_2). For probability measures with finite second moment and matching first moments, it is given by a supremum over functions whose first derivative is (1)-Lipschitz or, equivalently, whose second derivative is bounded by (1). Recent work places this metric at the intersection of smooth weak convergence, optimal transport, martingale transport, and quantitative limit theorems, and establishes sharp inequalities relating it to the quadratic Wasserstein distance [2401.04879], [2511.00232].

1. Definition and admissible test classes

For probability measures (\mu,\nu) on (\mathbb R), the general Zolotarev distance of order (p>0) is defined by writing (p=\ell+\delta), where (\ell=\lfloor p\rfloor) and (\delta=p-\ell\in(0,1]), and setting
[
|f|{\Lambda_p}=\sup{x\neq y}\frac{|f^{(\ell)}(x)-f^{(\ell)}(y)|}{|x-y|^\delta},
\qquad
\Lambda_p={f:\mathbb R\to\mathbb R,\ f\in C^\ell(\mathbb R),\ |f|{\Lambda_p}\le 1},
]
followed by
[
\zeta_p(\mu,\nu)=\sup{f\in\Lambda_p}\left|\int_\mathbb R f\,d\mu-\int_\mathbb R f\,d\nu\right|.
]
When (p=2), one has (\ell=1), (\delta=1), and
[
H_2:=\Lambda_2={f\in C^1(\mathbb R): |f'(x)-f'(y)|\le |x-y| \text{ for all }x,y},
]
so that
[
\zeta_2(\mu,\nu)=\sup_{f\in H_2}|E[f(X)]-E[f(Y)]|.
]
Equivalently one may require (f\in C^2(\mathbb R)) with (|f''(x)|\le 1) for all (x) [2401.04879].

In the multivariate setting, for (\mu,\nu\in P_2(\mathbb R^d)) with the same barycentre ([\mu]=[\nu]), the second Zolotarev distance is written
[
Z_2(\mu,\nu):=\sup\left{\int u\,d(\nu-\mu): u\in C^{1,1}(\mathbb R^d),\ \mathrm{Lip}(\nabla u)\le 1\right}.
]
Equivalently one may impose the pointwise Hessian constraint
[
D^2u(x)\preceq I \qquad \text{(a.e.)}
]
meaning that the operator norm (|D^2u(x)|_{\mathrm{op}}\le 1). A two-point Taylor argument yields an equivalent three-point inequality:
[
u(y)+\langle \nabla u(y),z-y\rangle-[u(x)+\langle \nabla u(x),z-x\rangle]
\le \frac12\bigl(|z-x|^2+|z-y|^2\bigr)
]
for all (x,y,z\in\mathbb R^d) [2511.00232].

A related multivariate formulation uses smooth test functions (u\in\mathscr U_2), where (\mathscr U_2) is the collection of all (C^\infty)-smooth functions (u:\mathbb R^d\to\mathbb R) satisfying (\sup_x|D^\alpha u(x)|\le 1) for every multi-index (\alpha) with (|\alpha|=2), together with (D^\alpha u(0)=0) for all (|\alpha|\le 1). Under the requirement that the first moments of (\mu) and (\nu) agree,
[
\zeta_2(\mu,\nu)=\sup_{u\in\mathscr U_2}\left|\int u\,d\mu-\int u\,d\nu\right|
]
[2506.17745].

2. Equivalent formulations and structural properties

On the real line, the second-order distance admits several equivalent representations. For a bounded signed Borel measure (M) on (\mathbb R), define
[
F_M(x)=M((-\infty,x]),
\qquad
F_{M,2}(x):=\int_{y\le x}(x-y)\,dM(y)=\int_{y\ge x}(y-x)\,dM(y).
]
Then
[
\zeta_2(M)=\int_{\mathbb R}|F_{M,2}(x)|\,dx,
]
and also
[
\zeta_2(M)=\sup_{|g''|\infty\le 1}\int g\,dM.
]
These formulas are stated on the subspace
[
M{2,1}:=\left{M\in\mathcal M:\int x^k\,dM(x)=0,\ k=0,1\right}
]
where the relevant versions of the Zolotarev norm coincide and define a genuine norm [2210.04060].

The metric properties recorded in the one-dimensional literature are standard: symmetry, triangle inequality, and absolute homogeneity. On (M_{2,1}), (\zeta_2) is a norm; more generally it is an ([0,\infty])-valued quasinorm on the space of bounded signed Borel measures. It is also translation invariant and obeys the scaling relation
[
\zeta_2(M(\lambda\cdot))=|\lambda|^2\,\zeta_2(M).
]
For probability laws with finite second moment, (\zeta_2) metrizes weak convergence plus convergence of second moments; in particular,
[
\zeta_2(\mu_n,\mu)\to 0
\quad\Rightarrow\quad
\mu_n\Rightarrow \mu
\ \text{ and }\
\int x^2\,d\mu_n\to \int x^2\,d\mu
]
[2401.04879], [2210.04060].

Several comparison statements place (\zeta_2) among other classical distances. For (01), Rio’s moment-comparison bound yields
[
W_p(\mu,\nu)\le c_p[\zeta_p(\mu,\nu)]^{1/p},
]
and in particular
[
W_2(\mu,\nu)\le c_2[\zeta_2(\mu,\nu)]^{1/2}.
]
On the real line one also has
[
\zeta_2(M)\le \max{\zeta_1(M),W_1(M)}\le v_2(M),
]
where (v_2(M)) is the second absolute moment, and
[
|M|_K\le \sqrt{\zeta_2(M)\,v_3(M)}
]
for the Kolmogorov distance (|\cdot|_K) when (v_3(M)<\infty) [2401.04879], [2210.04060].

3. Hessian Kantorovich–Rubinstein duality

A central development in the multivariate theory is a new Kantorovich–Rubinstein duality principle for the Hessian. For (\mu,\nu\in P_2(\mathbb R^d)) with ([\mu]=[\nu]), one introduces the set (\Sigma(\mu,\nu)) of probability measures (\pi) on ((\mathbb R^d)^3) whose ((x,y))-marginal is a coupling of (\mu) and (\nu) and whose third coordinate (z) satisfies the equilibrium constraints
[
\int\langle z-x,\Phi(x)\rangle\,d\pi=0
\qquad\text{and}\qquad
\int\langle z-y,\Psi(y)\rangle\,d\pi=0
]
for all suitable test-fields (\Phi,\Psi). Equivalently, (\pi) can be disintegrated so that both (\pi_{1,3}) and (\pi_{2,3}) are martingale couplings [2511.00232].

The duality theorem then states
[
Z_2(\mu,\nu)=\inf_{\pi\in\Sigma(\mu,\nu)}\int \frac12\bigl(|z-x|^2+|z-y|^2\bigr)\,d\pi(x,y,z).
]
Moreover, the infimum is attained, the supremum over (u\in C^{1,1}) with (\mathrm{Lip}(\nabla u)\le 1) is attained, and there is an explicit matching characterization: if (u) and (\pi) solve the two problems, then (\pi) concentrates on
[
z=z_u(x,y)=\frac{x+y}{2}+\frac{\nabla u(y)-\nabla u(x)}{2},
]
and on the set where the three-point inequality becomes an equality [2511.00232].

The proof strategy described for this result starts from the three-point inequality, which implies
[
\int u\,d(\nu-\mu)\le \frac12\int \bigl(|z-x|^2+|z-y|^2\bigr)\,d\pi
]
for any (\pi) satisfying the equilibrium conditions. A refined convex-analysis / martingale-transport argument then yields equality of supremum and infimum. This transport representation gives a “primal” formulation for a distance originally presented in dual form, and it is the basis for the sharp inequalities with (W_2) proved in the same work [2511.00232].

4. Sharp comparison with Wasserstein distance

The multivariate theory now contains two complementary sharp inequalities comparing the second Zolotarev distance with the quadratic Wasserstein distance. The first is a lower bound valid for all (\mu,\nu\in P_2(\mathbb R^d)):
[
Z_2(\mu,\nu)\ge \frac14\,W_2^2(\mu,\nu).
]
It follows immediately from the algebraic inequality
[
\frac12\bigl(|z-x|^2+|z-y|^2\bigr)\ge \frac14|x-y|^2
]
inserted into the infimum representation. The constant (1/4) is optimal, for example by two-point or three-point Dirac examples, and equality of both sides occurs only in the trivial case (\mu=\nu) [2511.00232].

The reverse bound requires variance information. If (\mu) and (\nu) are centred probabilities with variances (\mathrm{Var}(\mu)=\sigma_\mu^2) and (\mathrm{Var}(\nu)=\sigma_\nu^2), then
[
Z_2(\mu,\nu)\le \frac12(\sigma_\mu+\sigma_\nu)\,W_2(\mu,\nu).
]
The same work states that
[
h(a,b):=\sup_{\sigma_\mu\le a,\sigma_\nu\le b}\frac{Z_2}{W_2}=\frac12(a+b).
]
A slightly weaker corollary is
[
Z_2(\mu,\nu)\le \sqrt{\frac{\mathrm{Var}\,\mu+\mathrm{Var}\,\nu}{2}}\,W_2(\mu,\nu),
]
strict unless (\mu=\nu) [2511.00232].

The proof of the upper bound uses a refined version of the Kantorovich–Rubinstein lemma, described as the “magic” identity
[
Z_2(\mu,\nu)=\int \frac12\langle x,\nabla u(x)\rangle\,d(\nu-\mu)(x),
]
valid at a maximizer (u), together with an optimal (W_2)-plan and four Cauchy–Schwarz steps. Equality in the upper bound holds if and only if (\nu) is a pure dilation of (\mu) about their common barycentre, namely (\nu=T_\lambda{}#\mu) with (T\lambda(x)=\lambda x) and (\lambda\ge 1), or conversely (\lambda\le 1) if (\sigma_\nu\le \sigma_\mu) [2511.00232].

These statements complement the older comparison
[
W_2(\mu,\nu)\le c_2[\zeta_2(\mu,\nu)]^{1/2},
]
which already placed (\zeta_2) above (W_2) at the level of moments. The sharp lower and upper bounds show a more rigid relationship in (P_2(\mathbb R^d)): up to the universal constant (1/4), the two distances are locked together from below, and under variance constraints they differ by at most a factor (\frac12(\sigma_\mu+\sigma_\nu)) [2401.04879], [2511.00232].

5. High-dimensional reduction and probabilistic applications

A quantitative Cramér–Wold theorem gives an upper bound for multivariate Zolotarev distance in terms of one-dimensional projections. If (1 \[
\mathfrak P_q(b)=\left\{\pi:\int |x|^q\,d\pi(x)\le b^q\right\}
\]
with equal mixed moments up to order \(p-1\), then there is an absolute constant \(c\), independent of \(d,p,q,b\), such that
\[
\zeta_p(\mu,\nu)\le (c\,d)^p\,b^{\,1-\beta}\,
\Bigl(\sup_{|\theta|=1}\zeta_p(\mu_\theta,\nu_\theta)\Bigr)^\beta,
\qquad
\beta=\frac{2}{2+d\,\frac{q}{p(q-p)}}.
\]
Specializing to \(p=2\), if \(\mu,\nu\in\mathfrak P_q(b)\), \(q>2), have the same barycenter, then
[
\zeta_2(\mu,\nu)\le (c\,d)^2\,b^{\,1-\beta}
\Bigl(\sup_{|\theta|=1}\zeta_2(\mu_\theta,\nu_\theta)\Bigr)^\beta,
\qquad
\beta=\frac{2}{2+\tfrac{d\,q}{2(q-2)}}.
]
If (\mu,\nu) are supported in the Euclidean ball of radius (b), one may take (q\to\infty), giving
[
\zeta_2(\mu,\nu)\le (c\,d)^2
\Bigl(\sup_{|\theta|=1}\zeta_2(\mu_\theta,\nu_\theta)\Bigr)^{4/(4+d)}.
]
The same paper records that a one-dimensional optimal rate (O(n^{-1})) in (\zeta_2) for normalized sums passes to a multivariate rate
[
O!\bigl(n^{-\,2/(4+d)}\bigr)
]
via this bound [2506.17745].

In stochastic-process applications, the second Zolotarev distance supplies quantitative CLT rates under low moment assumptions. For a supercritical branching process in an independent and identically distributed environment, with (X_i=\log m_{i-1}), (\mu=E[X_i]), (\sigma^2=\mathrm{Var}(X_i)), and
[
E|X-\mu|^{2+\delta}<\infty
\qquad\text{for some }\delta\in(0,1),
]
Theorem 2.7 gives that for every (r\in[\delta,2]),
[
\zeta_r!\left(\frac{\log Z_n-n\mu}{\sigma\sqrt n},\mathcal N(0,1)\right)\le C\,n^{-\delta/2}.
]
In particular, at (r=2),
[
\zeta_2!\left(\frac{\log Z_n-n\mu}{\sigma\sqrt n},\mathcal N(0,1)\right)\le C\,n^{-\delta/2},
]
where (C>0) depends only on the moment constants but not on (n) [2401.04879].

On the real line, convolution inequalities show a stability under smoothing. For any (M_1,M_2\in\mathcal M),
[
\zeta_2(M_1*M_2)\le \zeta_1(M_1)\,|M_2|{TV}
+|M_1|{TV}\,\zeta_1(M_2),
]
and for probability laws this is interpreted as showing that smoothing by convolution cannot increase the (\zeta_2)-distance more than linearly in the (W_1)-distance of the factors [2210.04060].

6. Notational scope and a common source of confusion

A common source of confusion is that the notation (Z_2) also appears in a different literature on rational approximation. For disjoint compact sets (E,F\subset\mathbb C), the Zolotarev number of order (2) is defined by
[
Z_2(E,F)=\inf_{r\in\mathcal R_{2,2}}
\frac{\sup_{z\in E}|r(z)|}{\inf_{z\in F}|r(z)|},
]
where (\mathcal R_{2,2}) denotes rational functions with numerator and denominator of degree at most (2). In that setting one has
[
Z_2(E,F)\ge h^{-2},
\qquad
h=\exp!\Bigl(\tfrac1{\mathrm{cap}(E,F)}\Bigr),
]
and, under the hypotheses of the main theorem,
[
Z_2(E,F)\le (2\,\mathrm{Rot}(E)+1)(2\,\mathrm{Rot}(F)+1)\,h^{-2}+\mathcal O(h^{-4}).
]
For symmetric real intervals (E=[-a,-b]), (F=[b,a]), the exact value is
[
Z_2(E,F)=\exp!\bigl(-2\pi K/K'\bigr)=q^2
]
with (q=e^{-\pi K/K'}) [1911.11882].

This suggests an overloaded notation rather than a single unified object. In the probabilistic literature covered above, the second Zolotarev distance is a metric or norm-like quantity on signed measures or probability laws, formulated through smooth test functions, integral representations, or transport duality. In the rational-approximation literature, (Z_2(E,F)) is defined through extremal rational functions on disjoint compact sets in (\mathbb C). The shared name reflects historical terminology, but the underlying domains, admissible objects, and applications are different [1911.11882].