A Note on Symmetric Random Vectors with an Application to Discrete Choice

This paper studies random vectors X featuring symmetric distributions in that i) the order of the random variables in X does not affect its distribution, or ii) the distribution of X is symmetric at zero. We derive a number of characterization results for such random vectors, thereby connecting the distributional symmetry to various notions of how (Euclidean) functions have been regarded as symmetric. In addition, we present results about the marginals and conditionals of symmetrically distributed random vectors, and apply some of our results to various transformations of random vectors, e.g., to sums or products of random variables, or in context of a choice probability system known from economic models of discrete choice