spreadout

Spreadout, in probability theory, refers to a property of a probability measure on a locally compact group that describes a smoothing effect under convolution. A probability measure μ on a group G is called spread-out if there exists an integer n ≥ 1 such that the n-fold convolution μ^{*n} is absolutely continuous with respect to the Haar measure m on G. Equivalently, μ^{*n} has a density with respect to m. On the real line with Lebesgue measure, this means there is some n for which μ^{*n} has a density f ∈ L^1 that is positive on a nonempty open set.

Spread-out measures are important in the study of random walks on groups because, after finitely many steps, the walk acquires a density with respect to Haar measure. This enables the use of analytical tools such as Fourier analysis and local limit theorems. The property also relates to irreducibility and aperiodicity in the sense that the walk can explore a substantial part of the space rather than remaining confined to a discrete subset.

Examples:

- A non-degenerate normal distribution on R is spread-out because its convolutions remain normal with increasing variance, producing a positive density on all of R.

- A uniform distribution on an interval is spread-out since its n-fold convolution becomes a smooth density supported on a larger interval.

- A measure supported on a lattice, or a pure point mass, is typically not spread-out, as all convolutions remain singular with respect to Lebesgue (or Haar) measure.

Spread-out is contrasted with measures that are purely discrete or supported on subgroups that prevent absolute continuity, illustrating a key distinction in the long-term behavior of associated stochastic processes.