On Geary’s theorem for the field of p-adic numbers

Abstract

Let ℚ_p, where p > 2, be a field of p-adic numbers. We consider two independent identically distributed random variables with values in ℚ_p and distribution μ with a continuous density. We prove that the sum and the squared difference of these random variables are independent if and only if μ is an idempotent distribution, i.e., a shift of the Haar distribution of a compact subgroup of the additive group of the field ℚ_p.