Existence of phase transition for heavy-tailed continuum percolation

We consider a continuum percolation model in $R^d$, where $d >= 2$. It is given by a homogeneous Poisson process of intensity \labda and independent radii random variables of common distribution of a positive random variable $r$. Let \labda_c be the critical intensity for the existence of infinite cluster. We provide conditions for positivity of \labda_c. In case Er^{2d−1} = \infty our result is new

Convergence rates in the local renewal theorem

We improve the results from [Lindvall, T., 1979. On coupling of discrete renewal processes. Z. Wahrscheinlichkeitsth. 48, 57–70] on subgeometric convergence rates in the local renewal theorem. The results are used in [Sapozhnikov, A., 2006. Subgeometric rates of convergence of f-ergodic Markov chains (submitted for publication)] to generalize the previous results on convergence rates for Markov chains [Tuominen, P., Tweedie, R.L., 1994. Subgeometric rates of convergence of f-ergodic Markov chains. Adv. Appl. Probab. 26, 775–798]