arXiv subjects

Philip Weissmann

Publications and source records attributed to Philip Weissmann.

5 recordsLinked to original sources

arXiv2019

Completely asymmetric stable processes conditioned to avoid an interval

In the recent article D\"oring et al. [4] the authors conditioned a stable process with two-sided jumps to avoid an interval. As usual the strategy was to find an invariant function for the process killed on entering the interval and to show that the corresponding h-transformed process is indeed the process conditioned to avoid an interval in a meaningful way. In the present article we consider the case of a completely asymmetric stable process. It turns out that the invariant function found in [4] does not exist or is not invariant but nonetheless, we will characterize the conditioned process as a Markov process.

math.PR↗

arXiv2019

Poisson limit for the number of cycles in a random permutation and the number of segregating sites

Consider a random permutation of $\{1, \ldots, \lfloor n^{t_2}\rfloor\}$ drawn according to the Ewens measure with parameter $t_1$ and let $K(n, t)$ denote the number of its cycles, where $t\equiv (t_1, t_2)\in\mathbb [0, 1]^2$. Next, consider a sample drawn from a large, neutral population of haploid individuals subject to mutation under the infinitely many sites model of Kimura whose genealogy is governed by Kingman's coalescent. Let $S(n, t)$ count the number of segregating sites in a sample of size $\lfloor n^{t_2}\rfloor$ when mutations arrive at rate $t_1/2$. We show that $K(n, (t_1/\log n, t_2))-1$ and $S(n, (t_1/\log n, t_2))$ induce unique random measures $\Pi_n^K$ and $\Pi_n^S,$ respectively, on the positive quadrant $[0, \infty)^2.$ Our main result is to show that in the coupling of $S(n, t)$ and $K(n, t)$ introduced in~\cite{Pitters2019} we have weak convergence as $n\to\infty$ \begin{align*} (\Pi_n^K, \Pi_n^S)\to_d (\Pi, \Pi), \end{align*} where $\Pi$ is a Poisson point process on $[0, \infty)^2$ of unit intensity. This complements the work in~\cite{Pitters2019} where it was shown that the process $\{(K(n, t), S(n, t)), t\in [0, 1]^2\},$ appropriately rescaled, converges weakly to the product of the same one-dimensional Brownian sheet.

math.PR↗

arXiv2018

Stable processes conditioned to hit an interval continuously from the outside

Conditioning stable L\'evy processes on zero probability events recently became a tractable subject since several explicit formulas emerged from a deep analysis using the Lamperti transformations for self-similar Markov processes. In this article we derive new harmonic functions and use them to explain how to condition stable processes to hit continuously a compact interval from the outside.

math.PR↗

arXiv2018

Levy Processes with finite variance conditioned to avoid an interval

Conditioning Markov processes to avoid a set is a classical problem that has been studied in many settings. In the present article we study the question if a Levy process can be conditioned to avoid an interval and, if so, the path behavior of the conditioned process. For Levy processes with finite second moments we show that conditioning is possible and identify the conditioned process as an h-transform of the original killed process. The h-transform is explicit in terms of successive overshoot distributions and is used to prove that the conditioned process diverges to plus infinity and minus infinity with positive probabilities.

math.PR↗

arXiv2018

Stable processes conditioned to avoid an interval

Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function. In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for $\alpha\geq 1$ go back to classical work of Blumenthal et al. and Port from the 1960s. For $\alpha<1$, we appeal to recent results centred around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all $\alpha\in (0,2)$. With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob $h$-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob $h$-transform, we verify that the conditioned process is transient.

math.PR↗