# Proximity in terms of characteristic functions for $n$-dimensional distributions

Let $$X\in \mathbb{R}^n$$ and $$Y\in \mathbb{R}^n$$ be random variables with characteristic functions $$\phi_X(t)$$ and $$\phi_Y(t)$$, respectively.

Suppose that \begin{align} \sup_{t \in \mathbb{R}^n} \frac{|\phi_X(t)-\phi_Y(t)|}{\|t\|} \le \epsilon. \quad (*) \end{align}

Question: Can we say something about how close the distributions of $$X$$ and $$Y$$ are in some metric over probability spaces? In other words, suppose that $$(*)$$ small what other distance would be small too.

For example, for $$n=1$$. There exists the following inequality \begin{align} L^2(P_X,P_Y) \le 2 \sup_{t \ge 0} \frac{|\phi_X(t)-\phi_Y(t)|}{t} \end{align} where $$L$$ is the L\'evy distance.

However, I was not able to locate similar inequality for $$n>1$$. One distance that I have taken a look at extensively is the Levy-Prokhorov metric. However, whenever I find anything it always depends on the derivatives of characteristic functions.

There is a good reason why you cannot get anything for the standard Levy-Prokhorov distance in high dimensions. Let's consider the uniform distribution on the sphere of radius $$R$$ in $$\mathbb R^3$$ and the mixture (with weight $$\frac 12$$ for each) of the uniform distributions on the spheres of radii $$R-r$$ and $$R+r$$ where $$R>2r$$. Then the difference of the characteristic functions is $$|F(R|t|)-\frac 12[F((R-r)|t|)+F((R+r)|t|)]|\le r^2|t|^2\max_{[(R-r)|t|,(R+r)|t|]}|F''|$$ where $$F(u)=\frac{\sin u}{u}$$. However, $$|F''(u)|\le \frac Cu$$ for $$u>0$$, so to have your condition, it suffices to ensure that $$\frac {Cr^2|t|^2}{R|t|/2}\le\varepsilon |t|,$$ i.e. $$2Cr^2/R\le\varepsilon$$, which still allows $$r$$ to grow without bound as $$R\to+\infty$$ for any fixed $$\varepsilon>0$$.
• I don't know many probabilistic distance in high dimensions. Do you have any suggestion on which distance is small if $\sup_{t \in \mathbb{R}^n} \frac{| \phi_X(t) - \phi_Y(t)|}{\|t\|}$ is small? – Boby Jan 14 at 13:31