Let $\pi_i:\mathbb{R}^k\rightarrow \mathbb{R}:x\mapsto x_i$ be the $i$-th projection map for each $1\leq i\leq k$.
Let $\{\mu_n\}_{n\in\mathbb{N}}$ be a sequence of Borel probability measures on $\mathbb{R}^k$.
My question is: If $\{(\pi_i)_*\mu_n\}_{n\in\mathbb{N}}$ is tight for every $1\leq i \leq k$, then is $\{\mu_n\}_{n\in\mathbb{N}}$ tight?
In other words: If marginals are tight, is the original sequence tight? How to prove this?
Thank you in advance!
Fix $\epsilon>0$. Since each coordinate is tight, we can find for each $i \in \{1,\ldots,k\}$ some $R_i>0$ such that
$$(\pi_i \circ \mu_n)([-R_i,R_i]^c) \leq \epsilon \quad \text{for all $n \in \mathbb{N}$.}$$
If we define $K := \times_{i=1}^k [-R_i,R_i]$, then $K \subseteq \mathbb{R}^k$ is a compact set. Morover,
$$\mu_n(K^c) = \mu_n(\{y \in \mathbb{R}^k; \exists i \in \{1,\ldots,k\}: |y_i|>R_i)$$
and so
$$\mu_n(K^c) \leq \sum_{i=1}^k\mu_n(\{y \in \mathbb{R}^k; |y_i|>R_i\}) = \sum_{i=1}^k (\pi_n \circ \mu_n)([-R_i,R_i]^c) \leq k \epsilon$$
for all $n \in \mathbb{N}$. As $k \in \mathbb{N}$ is a fixed number and $\epsilon>0$ is arbitrary, this proves the tightness of $(\mu_n)_{n \in \mathbb{N}}$.