Tight Markov chains and random compositions

For an ergodic Markov chain $\{X(t)\}$ on $\Bbb N$, with a stationary distribution $\pi$, let $T_n>0$ denote a hitting time for $[n]^c$, and let $X_n=X(T_n)$. Around 2005 Guy Louchard popularized a conjecture that, for $n\to \infty$, $T_n$ is almost Geometric($p$), $p=\pi([n]^c)$, $X_n$ is almost stationarily distributed on $[n]^c$, and that $X_n$ and $T_n$ are almost independent, if $p(n):=\sup_ip(i,[n]^c)\to 0$ exponentially fast. For the chains with $p(n) \to 0$ however slowly, and with $\sup_{i,j}\,\|p(i,\cdot)-p(j,\cdot)\|_{TV}<1$, we show that Louchard's conjecture is indeed true even for the hits of an arbitrary $S_n\subset\Bbb N$ with $\pi(S_n)\to 0$. More precisely, a sequence of $k$ consecutive hit locations paired with the time elapsed since a previous hit (for the first hit, since the starting moment) is approximated, within a total variation distance of order $k\,\sup_ip(i,S_n)$, by a $k$-long sequence of independent copies of $(\ell_n,t_n)$, where $\ell_n= \text{Geometric}\,(\pi(S_n))$, $t_n$ is distributed stationarily on $S_n$, and $\ell_n$ is independent of $t_n$. The two conditions are easily met by the Markov chains that arose in Louchard's studies as likely sharp approximations of two random compositions of a large integer $\nu$, a column-convex animal (cca) composition and a Carlitz (C) composition. We show that this approximation is indeed very sharp for most of the parts of the random compositions. Combining the two approximations in a tandem, we are able to determine the limiting distributions of $\mu=o(\ln\nu)$ and $\mu=o(\nu^{1/2})$ largest parts of the random cca composition and the random C-composition, respectively. (Submitted to Annals of Probability in August, 2009.

A Kuznets Curve for Recycling

The paper aims at extending the debate on Environmental Kuznets Curves to the case of non-renewable resources and to discuss the driving forces that might give rise to EKC's in this case. The paper at hand deviates from the standard EKC analysis in two ways: First, mostly EKC's are analyzed for flow variables. In this paper we argue that EKC's may very well arise for certain stock variables like minerals or waste. Second, most papers that provide a theoretical foundation for EKC's focus on assumptions like technological anomalies (e.g. increasing returns) or technological switches. We offer an alternative explanation by showing that EKC's might arise simply due to the combination of recycling and the rising scarcity of materials. It is shown that an EKC for non-renewables might emerge during the transition to the long-run balanced growth path. Whether or not an EKC arises depends e.g. on initial conditions, but also on preferences and technology. The assumptions made about the ability of recycling firms to internalize the in- terrelation between recycling decisions today and the future availability of recyclable waste matter with respect to the prerequisites for an EKC and the speed of conver- gence. Internalization furthermore implies that an economy can be caught in a poverty trap, i.e. it might not be able to converge to the long-run growth equilibrium if the initial endowment with resources and capital is too low.non-renewable resources, recycling, transitional growth, Environmental Kuznets Curve