arXiv subjects
Publications and source records attributed to Ron Doney.
Several terms in an asynptotic estimate for the renewal mass function ina discrete random walk which has positive mean and regularly varying right-hand tail are given. Similar results are given for the renewal density function in the absolutely continuous case.
If the step distribution in a renewal process has finite mean and regularly varying tail with index -α, 1<α<2, the first two terms in the asymptotic expansion of the renewal function have been known for many years. Here we show that, without making any additional assumptions, it is possible to give, in all cases except for α=3/2 , the exact asymptotic behaviour of the next term. In the case α=3/2 the result is exact to within a slowly varying correction. Similar results are shown to hold in the random walk case.
We establish two different, but related results for random walks in the domain of attraction of a stable law of index $α$. The first result is a local large deviation upper bound, valid for $α\in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $α\in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long standing problem, which dates back to the 1962 paper of Garsia and Lamperti [Comm. Math. Helv.] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Pacific J. Math.] for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790
According to the Wiener-Hopf factorization, the characteristic function $φ$ of any probability distribution $μ$ on $\mathbb{R}$ can be decomposed in a unique way as \[1-sφ(t)=[1-χ_-(s,it)][1-χ_+(s,it)]\,,\;\;\;|s|\le1,\,t\in\mathbb{R}\,,\] where $χ_-(e^{iu},it)$ and $χ_+(e^{iu},it)$ are the characteristic functions of possibly defective distributions in $\mathbb{Z}_+\times(-\infty,0)$ and $\mathbb{Z}_+\times[0,\infty)$, respectively. We prove that $μ$ can be characterized by the sole data of the upward factor $χ_+(s,it)$, $s\in[0,1)$, $t\in\mathbb{R}$ in many cases including the cases where: 1) $μ$ has some exponential moments; 2) the function $t\mapstoμ(t,\infty)$ is completely monotone on $(0,\infty)$; 3) the density of $μ$ on $[0,\infty)$ admits an analytic continuation on $\mathbb{R}$. We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: {\it Any probability measure $μ$ on $\mathbb{R}$ whose support is not included in $(-\infty,0)$ is determined by its convolution powers $μ^{*n}$, $n\ge1$ restricted to $[0,\infty)$}. We show that in many instances, the sole knowledge of $μ$ and $μ^{*2}$ restricted to $[0,\infty)$ is actually sufficient to determine $μ$. Then we investigate the analogous problem in the framework of infinitely divisible distributions.
We consider the passage time problem for Lévy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in.
We consider a transient random walk on $Z^d$ which is asymptotically stable, without centering, in a sense which allows different norming for each component. The paper is devoted to the asymptotics of the probability of the first return to the origin of such a random walk at time $n$.
We call a right-continuous increasing process $K_x$ a partial right inverse (PRI) of a given Lévy process $X$ if $X_{K_x}=x$ for at least all $x$ in some random interval $[0,ζ)$ of positive length. In this paper, we give a necessary and sufficient condition for the existence of a PRI in terms of the Lévy triplet.
Let $R_n=\max_{0\leq j\leq n}S_j-S_n$ be a random walk $S_n$ reflected in its maximum. Except in the trivial case when $P(X\ge0)=1$, $R_n$ will pass over a horizontal boundary of any height in a finite time, with probability 1. We extend this by giving necessary and sufficient conditions for finiteness of passage times of $R_n$ above certain curved (power law) boundaries, as well. The intuition that a degree of heaviness of the negative tail of the distribution of the increments of $S_n$ is necessary for passage of $R_n$ above a high level is correct in most, but not all, cases, as we show. Conditions are also given for the finiteness of the expected passage time of $R_n$ above linear and square root boundaries.