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$\gamma\geq0$,\alpha>1$\alpha>1$, K>0$K>0$ and an integer n_0 such$n_0$ such that E(|X_{n}(t_{2})-X_{n}(t_{1})|^{\gamma})\leq K|t_{2}-t_{1}|^{\alpha}, \forall n\geq n_0$$E(|X_{n}(t_{2})-X_{n}(t_{1})|^{\gamma})\leq K|t_{2}-t_{1}|^{\alpha}, \forall n \geq n_0$$ 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?
