arXiv subjects

R. A. Doney

Publications and source records attributed to R. A. Doney.

7 recordsLinked to original sources

arXiv2017

Cramér's Estimate for the Reflected Process Revisited

The reflected process of a random walk or Lévy process arises in many areas of applied probability, and a question of particular interest is how the tail of the distribution of the heights of the excursions away from zero behaves asymptotically. The Lévy analogue of this is the tail behaviour of the characteristic measure of the height of an excursion. Apparently the only case where this is known is when Cramér's condition hold. Here we establish the asymptotic behaviour for a large class of Lévy processes which have exponential moments but do not satisfy Cramér's condition. Our proof also applies in the Cramér case, and corrects a proof of this given in Doney and Maller [5].

math.PR↗

arXiv2017

The strong renewal theorem with infinite mean via local large deviations

A necessary and sufficient condition is established for an asymptotically stable renewal process to satisfy the strong renewal theorem. This result is valid for all alpha in (0, 1), thus completing a result for alpha in (1/2, 1) which was proved in the 1963 paper of Garsia and Lamperti [6]. This paper is superseded by arXiv:1612.07635.

math.PR↗

arXiv2010

The asymptotic behavior of densities related to the supremum of a stable process

If $X$ is a stable process of index $α\in(0,2)$ whose Lévy measure has density $cx^{-α-1}$ on $(0,\infty)$, and $S_1=\sup_{0 x)\backsim Aα^{-1}x^{-α}$ as $x\to\infty$ and $P(S_1\leq x)\backsim Bα^{-1}ρ^{-1}x^{αρ}$ as $x\downarrow0$. [Here $ρ=P(X_1>0)$ and $A$ and $B$ are known constants.] It is also known that $S_1$ has a continuous density, $m$ say. The main point of this note is to show that $m(x)\backsim Ax^{-(α+1)}$ as $x\to\infty$ and $m(x)\backsim Bx^{αρ-1}$ as $x\downarrow0$. Similar results are obtained for related densities.

math.PR↗

arXiv2007

A note on the supremum of a stable process

If $X$ is a spectrally positive stable process of index $α\in(1,2)$ whose Lévy measure has density $cx^{-α-1}$ on $(0,\infty),$ and $S_1=\sup_{0 x)\backsim cα^{-1}x^{-α}$ as $x\to\infty.$ It is also known that $S_1$has a continuous density, $s$ say. The point of this note is to show that $s(x)\backsim cx^{-(α+1)}$ as $x\to\infty.$

math.PR↗

arXiv2006

Overshoots and undershoots of Lévy processes

We obtain a new fluctuation identity for a general Lévy process giving a quintuple law describing the time of first passage, the time of the last maximum before first passage, the overshoot, the undershoot and the undershoot of the last maximum. With the help of this identity, we revisit the results of Klüppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801] concerning asymptotic overshoot distribution of a particular class of Lévy processes with semi-heavy tails and refine some of their main conclusions. In particular, we explain how different types of first passage contribute to the form of the asymptotic overshoot distribution established in the aforementioned paper. Applications in insurance mathematics are noted with emphasis on the case that the underlying Lévy process is spectrally one sided.

math.PR↗

arXiv2005

Cramer's estimate for a reflected Levy process

The natural analogue for a Levy process of Cramer's estimate for a reflected random walk is a statement about the exponential rate of decay of the tail of the characteristic measure of the height of an excursion above the minimum. We establish this estimate for any Levy process with finite negative mean which satisfies Cramer's condition, and give an explicit formula for the limiting constant. Just as in the random walk case, this leads to a Poisson limit theorem for the number of ``high excursions.''

math.PR↗

arXiv2004

Stochastic bounds for Levy processes

Using the Wiener-Hopf factorization, it is shown that it is possible to bound the path of an arbitrary Levy process above and below by the paths of two random walks. These walks have the same step distribution, but different random starting points. In principle, this allows one to deduce Levy process versions of many known results about the large-time behavior of random walks. This is illustrated by establishing a comprehensive theorem about Levy processes which converge to \infty in probability.

math.PR↗