We know about Kolmogorov Criterion for the tightness of a stochastic process X_n(t)$X_n(t)$
1.The sequence (X_{n}(0))_{n\geq0} $(X_{n}(0))_{n\geq0}$ is tight.
2.There exist constants \gamma\geq0,\alpha>1, K>0 $\gamma\geq0$,$\alpha>1$, $K>0$ and an integer n_0 $n_0$ such that E(|X_{n}(t_{2})-X_{n}(t_{1})|^{\gamma})\leq $$E(|X_{n}(t_{2})-X_{n}(t_{1})|^{\gamma})\leq K|t_{2}-t_{1}|^{\alpha}, \forall n\geq n_0 n \geq n_0$$ for all t_{1},t_{2}.$t_{1},t_{2}$.
My first question: what should the n_0 $n_0$ depend? Could it depend on the t_{1} $t_{1}$ and t_{2}?$t_{2}$?
My second question: Is there any other criterion for tightness with the parameter \alpha=1 $\alpha=1$ for the version of the moment condition?
