I am searching for the criterion stated above and also here: [The question about Kolmogorov tightness criterion](https://mathoverflow.net/questions/39098/the-question-about-kolmogorov-tightness-criterion).


It should state the following: If a sequence of stochastic processes $(X^n)$ fulfills:


$$\mathbb{E}[|X^n\_t-X^n\_{t'}|^p]\leq C|t-t'|^\alpha$$


then it is tight.


I don't know if my googling skills are just too bad, but I can't find any source for that. Thanks for any advice!



Kolmogorov tightness criterion for stochastic processes
I assume that you want to show tightness in a space such as $C([0,1])$, as was the case in the question you link to. In fact, the approach of this answer will show tightness in $C^\beta([0,1])$ for every $\beta \in (0, \frac{\alpha - 1}{p})$.


Firstly, note that the statement you write cannot be sufficient for tightness since if $X^n$ is a sequence of constant processes then your condition trivially holds. Such a sequence need not be tight. The extra condition in your linked question that $(X\_0^n)\_{n \geq 1}$ is a tight sequence in $\mathbb{R}$ prevents such counterexamples. 


The key point is that from the proof of Kolmogorov's Continuity Criterion one can derive control on Holder norms of your process. For $\gamma \in (0, \frac{\alpha - 1}{p})$, one has the bound
$$\mathbb{E}([X^n]\_\gamma^p) \leq C(p, \alpha, \gamma) \cdot C$$
where $C(p,\alpha,\gamma)$ is a constant depending only on $p, \alpha$ and $\gamma$ (but is independent of $n$) and $[\cdot]\_\gamma$ is the usual $\gamma$-Holder seminorm on $C^\gamma([0,1])$. See [this answer](https://mathoverflow.net/a/279085/87850) for a proof. 


Let $\|X\|\_\gamma = |X\_0| + [X]\_\gamma$ denote (a norm equivalent to) the usual $\gamma$-Holder norm. Fix here $\varepsilon > 0$. By tightness of $(X\_0^n)$ there is an $M\_1$ such that $$\sup\_n \mathbb{P}(|X\_0^n| > M\_1) \leq \varepsilon$$


Also, by Markov's inequality and our above control on the Holder seminorm, we have that for $M\_2$ sufficiently large,
$$\sup\_n \mathbb{P}([X^n]\_\gamma > M\_2) \lesssim M\_2^{-p} \leq \varepsilon.$$


Hence $$\sup\_n\mathbb{P}(\|X^n\|\_\gamma > M\_1 + M\_2) \leq \sup\_n\mathbb{P}(|X\_0^n| > M\_1) + \sup\_n \mathbb{P}([X^n]\_\gamma > M\_2) \lesssim \varepsilon.$$
Finally, by compactness of the embedding $C^\gamma([0,1]) \to C([0,1])$ the closed ball of radius $M\_1 + M\_2$ in $C^\gamma([0,1])$ is relatively compact in $C([0,1])$ so the above inequality yields tightness of your sequence in $C([0,1])$.


