Consider a sequence of probability densities on $\mathbb{R}^n$, $\{\mu_n\}\subset \mathcal{P}(\mathbb{R}^n)$. I am interested in showing that this sequence is tight, and I would like to know if there is a sufficient condition on the second moments
$$ M(\mu_n):=\int_{\mathbb{R}^n} x^2 \mu_n(x)dx $$
which guarantees tightness.
The second moments being (uniformly) bounded suffices by the Markov inequality.
Edit: as Nejimban commented, the weaker condition that the first absolute moments are bounded suffices.