Hello people, I'm needing some help here.

Is it correct to say that a family of probability measure with finite p-moment is tight?

I saw this result only for gaussians (for p = 2, Billingsley, Probability and Measure, 3ed, page 337).

For example, let Xn be a r.v with finite p-moment. So, by Chebychev inequality,

Pn({x : |x| > c) \leq E|Xn|^p / c^p < K/c^p.

Therefore, taking sup_n in both sides, for a large c,

Pn({x : |x| > c) < \epsilon

and so the family {Pn} is tight. Is it correct?

Thanks in advance.