Kolmogorov tightness criterion for stochastic processes

Question

I am searching for the criterion stated above and also here: 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!

Answer 1

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 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 $C([0,1])$.