Drawing perturbations from a fat-tailed distribution I am wanting to apply random perturbations to a dynamical system. I woul dlike these perturbations to come from a fait tailed distrubtuion. I thought of using a t-distribution with a low dof number (1?). I would like the values to values to fall between different values: for instance, can I have them fall mainly (95%) between -0.1 to 0.1. How would i generalize this to any value, say from -A to A, where A could be 0.001 to 0.2. I would like to draw 1000 samples.
 A: The Cauchy distribution in R (rcauchy, pcauchy, etc) gives you the ability to change the scale (something like variance) while maintaining the super-fat tails of the $t_1$ distribution. You can play with the scale parameter to find what value gives you the $0.025$ quantile at $-0.1$ like you want.
The theory behind this is that you’re stretching or compressing $t_1$ but not really changing the shape. Think about a norma distribution with different variance parameters. Sure, the curve is different, but the spirit is the same: a mound shape.
Be warned that the tails of a Cauchy distribution are really heavy, so heavy that Cauchy doesn’t even have an expected value!
A: A fat-tailed distribution with well-defined moments, amiable to statistical testing is the log-normal distribution. To quote from Wikipedia: 

Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences as well as medicine, economics and other fields, e.g. for energies, concentrations, lengths, financial returns and other amounts.

