Bounding the total variation distance between two measures from a given set

I have a distance on the space of probability measures on $$[0,2]$$. It is defined as such for two probability measures $$\mu_1$$ and $$\mu_2$$ :

$$d_p(\mu_1,\mu_2) := \sum_{k=0}^p ( \mathbb{E}[X_1 ^k]- \mathbb{E}[X_1 ^k] ) ^2$$ where $$X_1$$ and $$X_2$$ are random variables whose laws are $$\mu_1$$ and $$\mu_2$$ respectively.

Let $$d_{TV}$$ be the total variation distance on the same space. I'm considering distributions lying in a set $$\Sigma$$ with :

$$\Sigma = \{ \alpha U(s,t) + (1-\alpha)W_1(R) \lvert (s,t,R,\alpha) \in [0,\epsilon_1]^2 \times [0,\epsilon_2] \times [0,1] \}$$

where :

$$\epsilon_1, \epsilon_2 >0$$, $$U(s,t)$$ is the uniform distribution on $$[s,t]$$ and $$W_1(R)$$ is the semi-circular law of parameter $$R>0$$ and centered on 1.

I want to know if there exists a constant $$C>0$$ (possibly for $$p$$ big enough) such that

$$\forall (\mu_1,\mu_2) \in \Sigma^2, \ d_{TV}(\mu_1,\mu_2) \leqslant C d_p(\mu_1,\mu_2)$$.

Is is possible to possible to find such a constant when $$\mu_1$$ and $$\mu_2$$ are both uniform distributions for instance but I do not know if it's true in the general case. Do you have an idea or reference for this problem? Any suggestion is welcome!

Thanks.

• This is hopeless. For any $p$, One can find pairs of probability measures where any the first $p$ moments agree. – Anthony Quas Jan 29 at 21:59
• Yes, for general probability measures, this is impossible but here I am studying the possibility of such a control on some subset $\Sigma$ of the probability space. – YZ22 Jan 30 at 0:03
• Still hopeless unless your subset is basically finite dimensional. – Anthony Quas Jan 30 at 3:20
• I don't understand Anthony's objection. Doesn't the question state that the subset is finite dimensional? – Steve Jan 30 at 8:57
• Yes indeed Steve, the subset is not finite dimensional per se but is parametrized by a small number of parameters. I'm going to update its definition in order to be clearer. – YZ22 Jan 30 at 9:39