Let $(X_n)_{n\in\mathbb N}$ be a discrete time stochastic process taking values in a Banach space $E.$ Suppose there exist constants $C,\alpha,\beta>0$ such that $\mathbb E\|X_n-X_m\|^\alpha\leq C|m-n|^{1+\beta}$ for all $m,n\geq 1.$ Is it true that there exist a almost surely equal version of $(X_n)$ say $(Y_n)$ such that for all $\omega\in\Omega$, $\omega$ being the probability space, for some $C(\omega)\geq 0$ $\|Y_n(\omega)-Y_m(\omega)\|\leq C(\omega)|m-n|^{\gamma}$ where $0\leq \gamma <\frac{\beta}{\alpha}$? Basically, I am asking if the Kolmogrov's theorem for existence of Hölder continuous trajectories is still valid for discrete time stochastic process or not. I can see that even if it is true one would require a new proof. The proof for continuous time stochastic process seems to be breaking down.

The answer is no. E.g., let $X_n=n$ for all natural $n$. Let $C=1$ and $\alpha=1$, and take any $\beta\in(0,1)$ and any natural $m<n$. Then $$E|X_n-X_m|^\alpha=E|X_n-X_m|=n-m\le C|m-n|^{1+\beta}. $$ However, $$|X_n(\omega)-X_m(\omega)|=n-m>C(\omega)|m-n|^\gamma $$ for all $\omega\in\Omega$, all real $C(\omega)$, and all $\gamma\in[0,\beta/\alpha)$ if $n>2m$ and $m$ is large enough. So, $$|Y_n(\omega)-Y_m(\omega)|=n-m>C(\omega)|m-n|^\gamma $$ for almost all $\omega\in\Omega$, all real $C(\omega)$, and all $\gamma\in[0,\beta/\alpha)$ if $n>2m$ and $m$ is large enough.