Let $X\_k$ be i.i.d. Cauchy random variables with parameters $0,1$. For each $N$ define the process $Y\_N$ by $$Y\_N(t)=\frac{1}N\sum\_{k=1}^{\lfloor tN\rfloor}X\_k+\text{piecewise linear interpolation}.$$


Note that for each grid point, the sum of Cauchy random variables is another Cauchy random variable. I am interested in the convergence of this process. Is the limit continuous? Is it Hölder? We cannot apply Kolmogorov continuity criterion due to lack of moments.



Hölder continuity of process from Donsker like theorem with Cauchy random variables
Indeed this falls a bit outside of the standard theory. As mentioned in this MSE answer to [Does a random walk with infinite mean ever converge to anything?](https://math.stackexchange.com/questions/1577043/does-a-random-walk-with-infinite-mean-ever-converge-to-anything), in particular for Cauchy too in the second answer, one has to use the stable FCLT to get that the limit of this random walk is the [Cauchy process](https://en.wikipedia.org/wiki/Cauchy_process). For example, as mentioned in the comments from these notes ["Stochastic-process limits"](http://www.columbia.edu/%7Eww2040/jumps.html) by Whitt, we can use theorem 4.5.3.


