We know about Kolmogorov Criterion for the tightness of a stochastic process X_n(t)

1.The sequence (X_{n}(0))_{n\geq0} is tight.

2.There exist constants \gamma\geq0,\alpha>1, K>0 and an integer 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
for all t_{1},t_{2}.

My first question: what should the n_0 depend? Could it depend on the t_{1} and t_{2}?

My second question: Is there any other criterion for tightness with the parameter \alpha=1 for the version of the moment condition?