Do the sample coefficients of variation follow a specific distribution when a lot of samples are taken?

Question

I am measuring the coefficient of variation after a process takes place, and I do not know the population distribution. I would like to find the probability the coefficient of variation is smaller than a specific value. I know that if I were looking at the sample mean, I would be able to use the central limit theorem if it were large enough. I was wondering if I could use a similar method for the sample coefficients of variation, or whether the coefficients of variation follow a distribution when a large enough set of samples are taken.