# Hölder continuity for discrete time process

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 $$m. Then $$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.