Kolmogorov tightness criterion says that if $X\_N$ is a sequence of continuous process with $X\_N(0)=0$ and $E[[X\_N(t)-X\_N(s)|^p]\leq C\_p |t-s|^{1+\beta}$ then for all $\gamma\in (0,\beta/p)$ we have that the laws of $X\_N$ are tight on the Holder space $C^\gamma$.


There are some easy examples where the assumption is not necessary. First of all, if $X\_N(t)=Z \psi(t)$ for all $N$ where $Z$ is a random variable with no moments and $\psi$ is as smooth as you want, then $P(\|X\_N\|\_\gamma>K)\to 0$ uniformly in $N$ as $K\to\infty$ and we have tightness on any Holder space.


I am wondering how we get around the lack of moments to prove tightness. For example, suppose that for all pairs $(s,t)$ we have $\lim\_{K\to\infty} P(\{|X\_N(t)-X\_N(s)|>K|t-s|^\gamma\})=0$ uniformly in $N$, then can we conclude tightness on $C^{\gamma'}$ for any $\gamma'<\gamma$?


Generally how you prove KTC is through Garsia-Rodemich-Rumsey inequality but it is hard to handle without moments.



Version of Kolmogorov tightness criterion without moments
Unfortunately, this does not hold, in the sense that you cannot conclude tightness even for any particular $\gamma’ < \gamma$. This very simple modification of my example [here](https://mathoverflow.net/questions/449705/garsia-rodemich-rumsey-without-markov) is a counterexample (I’ve just replaced the $\gamma$ in the exponent with $\gamma’$):


For any fixed positive integer $n \geq 22$, let $X^n$ be defined as follows - uniformly at random pick a point $p$ in, $[\frac{1}{4}, \frac{3}{4}]$, and define


$$
X^n :=
\begin{cases}
0 & \text{on } [0, p], \\
n(t - p)^{\gamma’} & \text{on } (p, p + \frac{1}{n^{2/\gamma’}}], \\
\frac{1}{n} & \text{on } (p + \frac{1}{n^{2/\gamma’}}, 1]. \\
\end{cases}
$$


Then


$$\lim\_{K\to\infty} P(\{|X\_n(t)-X\_n(s)|>K|t-s|^\gamma\})=0$$


indeed uniformly in $n, s, t$ but $\| X^n \|\_{\gamma’} = n$ almost surely, which tends to $\infty$ and so tightness in Holder space is not possible.


I can do the calculation in full if needed, but morally the reason why this counterexample works is this:


The requested limit above can be shown to hold even if $X\_n$ were the process that was discontinuous at $p$ with a jump of size $\frac{1}{n}$. Of course we have assumed $X\_n$ continuous, but it follows that you could have $X\_n$ continuous but make the bump essentially as bad as you want in $\gamma’$ Holder norm and still have that limit hold.


I second the suggestion of Thomas Kojar to try to look at the particular process and see if we can come up with some criterion that fits your purpose.


