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Correction to: Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula

  • Xiao Fang
  • Qi-Man Shao
  • Lihu XuEmail author
Correction
  • 11 Downloads

1 Correction to: Probability Theory and Related Fields  https://doi.org/10.1007/s00440-018-0874-5

We write this note to correct [1, (6.9), (6.13), (7.1), (7.2)] because there was one term missed in [1, (6.9)]. To estimate this missed term, we need to add an extra condition to [1, Assumption 2.1]:

Assumption 2.1

\(g \in \mathcal C^3(\mathbb {R}^d,\mathbb {R}^d)\), and there exist \(\theta _0>0\) and \(\theta _1, \theta _2, \theta _3, \theta _3' \ge 0\) such that the conditions (2.3) and (2.4) in [1] hold, i.e.,
$$\begin{aligned} \langle u, \nabla _u g(x)\rangle \le -\theta _0 \left( 1+\theta _1 |x|^{\theta _2}\right) |u|^2, \ \ \ \ \ \quad&\forall \ u, x \in \mathbb {R}^d;\\ |\nabla _{u_1} \nabla _{u_2} g(x)| \le \theta _3 (1+ \theta _1 |x|)^{\theta _2-1} |u_1| |u_2|, \quad&\forall \ u_1, u_2, x \in \mathbb {R}^d. \end{aligned}$$
and additionally,
$$\begin{aligned} |\nabla _{u_1} \nabla _{u_2} \nabla _{u_3} g(x)| \le \theta _3' (1+|x|)^{\theta _2-2} |u_1| |u_2| |u_3|, \ \ \ \ \ \forall \ u_1, u_2, u_3, x \in \mathbb {R}^d; \end{aligned}$$
Under the above-strengthened Assumption 2.1, all the conclusions and examples in [1] still hold true, except that all the constants \(C_\theta \) therein will depend on the constants in the new assumption.
Before correcting [1, (6.9), (6.13), (7.1), (7.2)], let us recall some notations in [1], give the missed term, and prove an auxiliary lemma. Let \(u, u_{1}, u_{2} \in \mathbb {R}^{d}\), recall
$$\begin{aligned} \mathcal I^{x}_{u}(t)=\frac{1}{\sqrt{2} t} \int _{0}^{t} \langle \nabla _{u} X^{x}_{s}, \mathrm {d}B_{s}\rangle , \ \ \ \mathcal I^{x}_{u_{1},u_{2}}(t)=\mathcal I^{x}_{u_{1}}(t) \mathcal I^{x}_{u_{2}}(t)-D_{V_{2}} \mathcal I^{x}_{u_{1}}(t) \end{aligned}$$
with \(V_{i,t}=\int _{0}^{t} v_{i}(s) \mathrm {d}s\) and \(v_{i}(s)=\frac{1}{\sqrt{2} t} \nabla _{u_{i}} X^{x}_{s}\) for \(0 \le s \le t\) and \(i=1,2\), see [1, (5.12),(5.13)]. The missed term is defined by
$$\begin{aligned} \mathcal R^{x}_{u_{1},u_{2}}(t):=\nabla _{u_{2}} \nabla _{u_{1}}X^{x}_{t}-D_{V_{2}} (\nabla _{u_{1}} X^{x}_{t}). \end{aligned}$$

Lemma 0.1

We have
$$\begin{aligned} |\mathcal R_{u_1, u_2}^x(t)|\le & {} C_\theta |u_2| |u_1|,\\ |\nabla _{u_3} \mathcal R_{u_1, u_2}^x(t)|\le & {} C_\theta |u_3| |u_2| |u_1|,\\ |D_{V_3} \mathcal R_{u_2, u_1}^x(t)|\le & {} C_\theta |u_3| |u_2| |u_1|, \end{aligned}$$
for all \(u_1,u_2,u_3, x\in \mathbb {R}^d\).

Proof

The first bound follows immediately from [1, (5.7),(5.17)]. It is easy to check that \(\nabla _{u_3} \nabla _{u_2} \nabla _{u_1} X_t\) satisfies the equation
$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}t}\nabla _{u_3} \nabla _{u_2} \nabla _{u_1} X_t= \nabla g(X_t)\nabla _{u_3} \nabla _{u_2} \nabla _{u_1} X_t+\nabla ^2 g(X_t) \mathcal R_1(t)+\nabla ^3 g(X_t) \mathcal R_2(t), \end{aligned}$$
where \(\mathcal R_1\) and \(\mathcal R_2\) are terms about first and second-order derivatives of \(X_t\). By [1, (5.6), (5.7)], we have
$$\begin{aligned} |\mathcal R_1(t)| \le C_\theta |u_1| |u_2| |u_3|, \ \ \ |\mathcal R_2(t)| \le C_\theta |u_1| |u_2| |u_3|. \end{aligned}$$
Differentiating \(|\nabla _{u_3} \nabla _{u_2} \nabla _{u_1} X_t|^2\) with respect to t and using the above two bounds, we can prove by the same argument as showing [1, (5.7)]
$$\begin{aligned} |\nabla _{u_3} \nabla _{u_2} \nabla _{u_1} X_t| \le C_\theta |u_1| |u_2| |u_3|. \end{aligned}$$
Similarly, first finding the differential equations of \(\nabla _{u_3} D_{V_2} \nabla _{u_1} X_t\), \(D_{V_3} \nabla _{u_2} \nabla _{u_1} X_t\), \(D_{V_3} D_{V_2} \nabla _{u_1} X_t\), and then using the same argument as above, we get
$$\begin{aligned}&|\nabla _{u_3} D_{V_2} \nabla _{u_1} X_t| \le C_\theta |u_1| |u_2| |u_3|, \\&|D_{V_3} \nabla _{u_2} \nabla _{u_1} X_t| \le C_\theta |u_1| |u_2| |u_3|, \\&|D_{V_3} D_{V_2} \nabla _{u_1} X_t| \le C_\theta |u_1| |u_2| |u_3|. \end{aligned}$$
Collecting the previous estimates, we immediately obtain the other two estimates in the lemma. \(\square \)
Correction to [1, (6.9), (6.13)]: The original [1, (6.9), (6.13)] should be corrected as
$$\begin{aligned} \nabla _{u_{2}} \nabla _{u_{1}} \mathbb {E}[\phi (X^{x}_{t})]=\mathbb {E}[\nabla _{u_{1}} \phi (X^{x}_{t}) \mathcal I^{x}_{u_{2}}(t)]+\mathbb {E}[\nabla \phi (X^{x}_{t}) \mathcal R^{x}_{u_{1},u_{2}}(t)] \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} \nabla _{u_2} \nabla _{u_1} f(x)&=\int _0^{\infty } e^{- t} \mathbb {E}\left\{ \big [\nabla _{u_1} f(X_t^x)-\nabla _{u_1} h(X_t^x)\big ] \mathcal I_{u_2}^x(t)\right\} \mathrm {d}t \\&\quad +\int _0^{\infty } e^{- t} \mathbb {E}\left\{ \big [\nabla f(X_t^x)-\nabla h(X_t^x)\big ] \mathcal R_{u_2, u_1}^x(t)\right\} \mathrm {d}t, \end{aligned} \end{aligned}$$
for \(u_1, u_2, x \in \mathbb {R}^d\).

Proof

We have
$$\begin{aligned} \begin{aligned} \nabla _{u_{2}} \nabla _{u_{1}} \mathbb {E}[\phi (X^{x}_{t})]=\mathbb {E}\left[ \nabla ^{2} \phi (X^{x}_{t}) \nabla _{u_{2}} X^{x}_{t}\nabla _{u_{1}}X^{x}_{t}\right] +\mathbb {E}\left[ \nabla \phi (X^{x}_{t})\nabla _{u_{2}} \nabla _{u_{1}}X^{x}_{t}\right] , \end{aligned} \end{aligned}$$
By [1, (5.14), (5.9), (5.11)],
$$\begin{aligned} \begin{aligned} \mathbb {E}\left[ \nabla ^{2} \phi (X^{x}_{t}) \nabla _{u_{2}} X^{x}_{t}\nabla _{u_{1}}X^{x}_{t}\right]&=\mathbb {E}\left[ \nabla ^{2} \phi (X^{x}_{t}) D_{V_{2}} X^{x}_{t}\nabla _{u_{1}}X^{x}_{t}\right] \\&=\mathbb {E}\left[ D_{V_{2}}(\nabla \phi (X^{x}_{t})) \nabla _{u_{1}}X^{x}_{t}\right] \\&=\mathbb {E}\left[ D_{V_{2}}(\nabla \phi (X^{x}_{t}) \nabla _{u_{1}}X^{x}_{t})\right] - \mathbb {E}\left[ \nabla \phi (X^{x}_{t}) D_{V_{2}} (\nabla _{u_{1}}X^{x}_{t})\right] \\&=\mathbb {E}\left[ \nabla _{u_{1}} \phi (X^{x}_{t}) \mathcal I^{x}_{u_{2}}(t)\right] -\mathbb {E}\left[ \nabla \phi (X^{x}_{t}) D_{V_{2}} (\nabla _{u_{1}}X^{x}_{t})\right] . \end{aligned} \end{aligned}$$
Combining the above relations, we immediately obtain the first relation in the proposition. The second relation can immediately be obtained from the first one. \(\square \)
Correction to the proofs of (7.1) and (7.2) in [1]: The conclusions of (7.1) and (7.2) still hold under the strengthened Assumption 2.1., but we need to estimate the extra terms related to \(\mathcal R_{u_2, u_1}^x(t)\). From the second relation in the above proposition, we have
$$\begin{aligned} \begin{aligned} \left| \nabla _{u_2}\nabla _{u_1} f(x)\right|&\le \int _0^\infty e^{- t} \left| \mathbb {E}\left\{ \big [\nabla _{u_1} f(X_t^x)-\nabla _{u_1}h(X_t^x)\big ] \mathcal I_{u_2}^x(t)\right\} \right| \mathrm {d}t \\&\quad +\int _0^\infty e^{-t} \left| \mathbb {E}\left\{ \big [\nabla f(X_t^x)-\nabla h(X_t^x)\big ] \mathcal R_{u_1, u_2}^x(t)\right\} \right| \mathrm {d}t. \end{aligned} \end{aligned}$$
Since we have shown in the original proof that
$$\begin{aligned} \int _0^\infty e^{- t} \left| \mathbb {E}\left\{ \big [\nabla _{u_1} f(X_t^x)-\nabla _{u_1}h(X_t^x)\big ] \mathcal I_{u_2}^x(t)\right\} \right| \mathrm {d}t \le C_\theta \Vert \nabla h\Vert |u_1| |u_2|, \end{aligned}$$
it remains to bound the second integral. By [1, (5.7), (5.17)], we immediately obtain
$$\begin{aligned} \int _0^\infty e^{-t} \left| \mathbb {E}\left\{ \big [\nabla f(X_t^x)-\nabla h(X_t^x)\big ] \mathcal R_{u_1, u_2}^x(t)\right\} \right| \mathrm {d}t \le C_\theta \Vert \nabla h\Vert |u_1| |u_2|. \end{aligned}$$
Combining the previous three inequalities, we conclude that [1, (7.1)] still holds true.
To prove [1, (7.2)], we have
$$\begin{aligned} {\begin{aligned}&\nabla _{u_2}\nabla _{u_1} f(x+\varepsilon u)-\nabla _{u_2}\nabla _{u_1}f(x)\ = \ \int _0^{\infty } e^{- t} \Psi \mathrm {d}t\ + \int _0^{\infty } e^{- t} \Phi \mathrm {d}t = J_1+J_2, \end{aligned}} \end{aligned}$$
where
$$\begin{aligned} {\begin{aligned} \Psi&= \mathbb {E}\left\{ \big [ \nabla _{u_1} f(X_t^{x+\varepsilon u})-\nabla _{u_1} h(X_t^{x+\varepsilon u})\big ] \mathcal I_{u_2}^{x+\varepsilon u}(t)\right\} \\&\quad -\mathbb {E}\left\{ \big [ \nabla _{u_1} f(X_t^x)-\nabla _{u_1} h(X_t^x)\big ] \mathcal I_{u_2}^x(t)\right\} . \\ \Phi&= \mathbb {E}\left\{ \big [ \nabla f(X_t^{x+\varepsilon u})-\nabla h(X_t^{x+\varepsilon u})\big ] \mathcal R_{u_1, u_2}^{x+\varepsilon u}(t)\right\} \\&\quad -\mathbb {E}\left\{ \big [ \nabla f(X_t^x)-\nabla h(X_t^x)\big ] \mathcal R_{u_1,u_2}^x(t)\right\} . \\ \end{aligned}} \end{aligned}$$
We have shown in the original proof that
$$\begin{aligned} \left| J_1\right| \le C_{\theta } ||\nabla h|| |\varepsilon | \left( |\log |\varepsilon || \vee 1\right) |u_{1}||u_{2}|. \end{aligned}$$
We prove below that
$$\begin{aligned} \left| J_2\right| \le C_{\theta } ||\nabla h|| |\varepsilon | |u_{1}||u_{2}| |u|. \end{aligned}$$
Combining the estimates of \(J_1\) and \(J_2\), we immediately get that [1, (7.2)] still holds true.
Let us show the above bound about \(J_2\). Write
$$\begin{aligned} J_{2} =J_{2,1}+J_{2,2}, \end{aligned}$$
with
$$\begin{aligned} J_{2,1}= & {} \int _0^{\infty } e^{- t} \mathbb {E}\left\{ \big [ \nabla f(X_t^{x+\varepsilon u})-\nabla h(X_t^{x+\varepsilon u})\big ] \big [\mathcal R_{u_1, u_2}^{x+\varepsilon u}(t)-\mathcal R_{u_1, u_2}^{x}(t)\big ]\right\} \mathrm {d}t,\\ J_{2,2}= & {} \int _0^{\infty } e^{- t} \mathbb {E}\left\{ \big [ \nabla f(X_t^{x+\varepsilon u})-\nabla h(X_t^{x+\varepsilon u})-\nabla f(X_t^x)+\nabla h(X_t^x)\big ] \mathcal R_{u_1, u_2}^{x}(t)\right\} \mathrm {d}t, \end{aligned}$$
For \(J_{2,1}\), observe
$$\begin{aligned} \begin{aligned} J_{2,1}&= \varepsilon \int _0^{\infty } e^{- t} \int _{0}^{1} \mathbb {E}\left\{ [\nabla f(X_t^{x+\varepsilon u})-\nabla h(X_t^{x+\varepsilon u})] \nabla _{u}\mathcal R_{u_1, u_2}^{x+s \varepsilon u}(t)\right\} \mathrm {d}s \mathrm {d}t, \end{aligned} \end{aligned}$$
which, together with Lemma 0.1, immediately gives
$$\begin{aligned} |J_{2,1}| \le C_{\theta }|\varepsilon | (\Vert \nabla f\Vert +\Vert \nabla h\Vert ) \le C_{\theta }|\varepsilon | \Vert \nabla h\Vert |u| |u_1| |u_2|. \end{aligned}$$
For \(J_{2,2}\), we have
$$\begin{aligned} \begin{aligned} J_{2,2}&= \varepsilon \int _0^{\infty } e^{- t} \int _{0}^{1} \mathbb {E}\left\{ \nabla [\nabla f(X_t^{x+s\varepsilon u})-\nabla h(X_t^{x+s\varepsilon u})] \nabla _{u} X_{t}^{x+s \varepsilon u} \mathcal R_{u_1, u_2}^{x}(t)\right\} \mathrm {d}s \mathrm {d}t \\&= \varepsilon \int _0^{\infty } e^{- t} \int _{0}^{1} \mathbb {E}\left\{ \nabla [\nabla f(X_t^{x+s\varepsilon u})-\nabla h(X_t^{x+s\varepsilon u})] D_{V} X_{t}^{x+s \varepsilon u} \mathcal R_{u_1, u_2}^{x}(t)\right\} \mathrm {d}s \mathrm {d}t \\&= \varepsilon \int _0^{\infty } e^{- t} \int _{0}^{1} \mathbb {E}\left\{ D_V [\nabla f(X_t^{x+s\varepsilon u})-\nabla h(X_t^{x+s\varepsilon u})] \mathcal R_{u_1, u_2}^{x}(t)\right\} \mathrm {d}s \mathrm {d}t \\&= \varepsilon (J_{2,2,1}-J_{2,2,2}) \end{aligned} \end{aligned}$$
where the last equality is by [1, (5.14), (5.9), (5.11)] and
$$\begin{aligned} J_{2,2,1}= & {} \int _0^{\infty } e^{- t} \int _{0}^{1} \mathbb {E}\left\{ [\nabla f(X_t^{x+s\varepsilon u})-\nabla h(X_t^{x+s\varepsilon u})] \mathcal R_{u_1, u_2}^{x}(t) \mathcal I^{x+s\varepsilon u}_u(t)\right\} \mathrm {d}s \mathrm {d}t,\\ J_{2,2,2}= & {} \int _0^{\infty } e^{- t} \int _{0}^{1} \mathbb {E}\left\{ \left[ \nabla f(X_t^{x+s\varepsilon u})-\nabla h(X_t^{x+s\varepsilon u})\right] D_V \mathcal R_{u_1, u_2}^{x}(t)\right\} \mathrm {d}s \mathrm {d}t. \end{aligned}$$
By Lemma 0.1 and [1, (5.18)],
$$\begin{aligned} |J_{2,2,2}|\le & {} C_\theta (\Vert \nabla f\Vert +\Vert \nabla h\Vert ) |u_1| |u_2| |u| \le C_\theta \Vert \nabla h\Vert |u_1| |u_2| |u|,\\ |J_{2,2,1}|\le & {} C_\theta (\Vert \nabla f\Vert +\Vert \nabla h\Vert ) |u_1| |u_2|\int _0^{\infty } e^{- t} \int _{0}^{1} \mathbb {E}\left[ |\mathcal I^{x+s\varepsilon u}_u(t)|\right] \mathrm {d}s \mathrm {d}t \\\le & {} C_\theta \Vert \nabla h\Vert |u_1| |u_2| |u|. \end{aligned}$$
Combining the estimates above, we immediately obtain the bound of \(J_2\). \(\square \)

Due to the new assumption on g, [1, Remark 3.2] should be revised as

Remark 3.2

Gorham et. al. (see [18] in [1]) recently put forward a method to measure sample quality with diffusions by a Stein discrepancy, in which the same Stein equation as (3.1) has to be considered. Under the assumption that g is third-order differentiable, they used the Bismut–Elworthy–Li formula (see [16] in [1]), together with smooth convolution and interpolation techniques, to prove a bound on the first, second, and \((3-\epsilon )\)th derivative of f for \(\epsilon >0\). They can also obtain the bound (3.4) by their approach (personal communication (see [24] in [1]) after their reading our paper on ArXiv) together with a limiting argument.

Notes

Acknowledgements

We thank Jim Dai and James Thompson for pointing out the errors.

Reference

  1. 1.
    Fang, X., Shao, Q.M., Xu, L.: Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula. Probab. Theory Relat. Fields (2019) (to appear) Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of StatisticsThe Chinese University of Hong KongShatinHong Kong
  2. 2.Department of Mathematics, Faculty of Science and TechnologyUniversity of MacauTaipa, MacauChina
  3. 3.Zhuhai UM Science and Technology Research InstituteZhuhaiChina

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