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

• Xiao Fang
• Qi-Man Shao
• Lihu Xu
Correction

## 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}
\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