# Sequential compactness of a sequence of curves of Borel probability measures

$$%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator*{\supp}{supp} \newcommand{\bR}{\mathbb{R}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bD}{\mathbb{D}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\cA}{\mathcal{A}} \newcommand{\cB}{\mathcal{B}} \newcommand{\cD}{\mathcal{D}} \newcommand{\cE}{\mathcal{E}} \newcommand{\cF}{\mathcal{F}} \newcommand{\cG}{\mathcal{G}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\sD}{\mathscr{D}} \newcommand{\sE}{\mathscr{E}} \newcommand{\sG}{\mathscr{G}} \newcommand{\sH}{\mathscr{H}} \newcommand{\sK}{\mathscr{K}} \newcommand{\sP}{\mathscr{P}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\eps}{\varepsilon} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\andd}{\quad \text{and} \quad} \newcommand{\qtext}{\quad\text} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%$$ Let $$\sP_2 (\bR^d)$$ be the space of Borel probability measure on $$\bR^d$$ with finite second moment.

• We denote by $$M_2 (\nu) := \int_{\bR^d} |x|^2 \diff \nu (x)$$ the second moment of $$\nu \in \sP_2 (\bR^d)$$. We denote by $$W_2$$ the $$2$$-Wasserstein metric on $$\sP_2 (\bR^d)$$.

• We denote by $$\rightharpoonup$$ the weak convergence in $$\sP_2 (\bR^d)$$, i.e., $$\nu_n \rightharpoonup \nu$$ in $$\sP_2 (\bR^d)$$ if and only if $$\int_{\bR^d} f \diff \nu_n \to \int_{\bR^d} f \diff \nu$$ for every $$f \in C_b (\bR^d)$$. Correspondingly, we denote by $$\tau_{\mathrm w}$$ the topology that $$\rightharpoonup$$ induces on $$\sP_2 (\bR^d)$$.

For each $$n \in \bN^*$$ and $$t \in [0, 1]$$, let $$\mu^n_t \in \sP_2 (\bR^d)$$. We fix $$\alpha \in (0, 1)$$ and assume that $$$$\sup_{t \in [0, 1]} \sup_{n \in \bN^*} M_2 (\mu^n_t) + \sup_{n \in \bN^*} \sup_{\substack{s,t \in [0, 1] \\ s \neq t}} \frac{W_2 (\mu^n_t, \mu^n_s)}{|t-s|^\alpha} < \infty.$$$$

Then for each $$t \in [0, 1]$$, the sequence $$(\mu^n_t)_{n \in \bN^*}$$ has compact closure in $$\tau_{\mathrm w}$$.

Are there $$(\mu_t)_{t \in [0, 1]} \subset \sP_2 (\bR^d)$$ and a sub-sequence $$(n_k)_{k \in \bN^*}$$ such that for each $$t \in [0, 1]$$: $$$$\mu^{n_k}_t \rightharpoonup \mu_t \quad \text{as} \quad k \to \infty.$$$$

• Are you assuming some continuity for $\mu_{t}$ in $t$?
– Asaf
Commented Jun 9 at 6:47
• @Asaf I am willing to assume such a continuity condition. Please see my edit. Commented Jun 9 at 6:56
• Errr... Isn't this an immediate application af the Arzelà-Ascoli theorem? Commented Jun 20 at 16:27
• @Akira so, is this Arzelà-Ascoli in the end, or not? I'm pretty damn sure it is, so I'm just checking on any progress here? Commented Jul 1 at 20:42
• @leomonsaingeon you are right! I am dumb... Commented Jul 5 at 8:17

Let $$T \subset [0,1]$$ be countable and dense. Using a diagonal sequence argument, one can find a subsequence such that $$\mu_n^{n_k} \rightharpoonup \mu_t$$ for all $$t \in T$$. Using your continuity assumption, $$T \ni t \mapsto \mu_t$$ should be Hölder continuous. Consequently, it should be possible to extend it to all of $$[0,1]$$ and to conclude with the desired convergence.