Let $P:\mathbb{R}\to \mathbb{R}:= P(x)=\sum_{j=1}^{d}a_jx^j $ be a given non-zero polynomial of degree $d\ge 2.$
(1) Let $\{P(X_n)\}$ be a sequence of uniformly tight random variables. N.B. (Uniformly) tight and stochastically bounded means the same thing. From here, can we in general infer that $\{X_n\}$ is also uniformly tight? If $P^{-1}$ exists and is unique, then it'll be continuous and so it'll map uniformly tight sequence to the same, but does the same argument hold for a general polynomial?
(2) Let $\alpha >0, \text{ and } n^{\alpha}P(X_n)=O_P(1)$ (tight). Does this also mean $n^{\alpha}{X_n}^{j}=O_P(1) \text{ (tight) } \forall 1\le j \le d$ ?
It seems to me that both, especially number (1) is true, but finding the precise argument seems to be difficult for me.