Precise multivariate tail asymptotics for the invariant measure

Establish a precise limiting asymptotic for the directional tail masses of the invariant Radon measure in the multivariate critical affine stochastic recursion, by obtaining the exact conditioned central and local limit theorems for the associated Markov random walks.

Background

For the critical multidimensional affine stochastic recursion, the paper proves two-sided logarithmic bounds for the directional mass of multiplicative intervals, namely bounds proportional to $\log(b/a)$ for directions in the support of the stationary measure on the positive sphere. In one dimension and for similarity matrices, sharper results establish an exact limit equal to a constant multiple of the Haar measure $ds/s$ on the multiplicative group.

The unresolved issue is to obtain the corresponding exact limit in the general multivariate setting. The authors identify exact asymptotics in conditioned central and local limit theorems for Markov random walks as the missing ingredient; these estimates are described as currently unavailable.