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Given a two-sided real-valued Levy process $(X_t)_{t in mathbb{R}}$, define processes $(L_t)_{t in mathbb{R}}$ and $(M_t)_{t in mathbb{R}}$ by $L_t := sup{h in mathbb{R} : h - alpha(t-s) le X_s text{ for all } s le t} = inf{X_s + alpha(t-s) : s le t}$, $t in mathbb{R}$, and $M_t := sup { h in mathbb{R} : h - alpha|t-s| leq X_s text{ for all } s in mathbb{R} } = inf {X_s + alpha |t-s| : s in mathbb{R}}$, $t in mathbb{R}$. The corresponding contact sets are the random sets $mathcal{H}_alpha := { t in mathbb{R} : X_{t}wedge X_{t-} = L_t}$ and $mathcal{Z}_alpha := { t in mathbb{R} : X_{t}wedge X_{t-} = M_t}$. For a fixed $alpha>mathbb{E}[X_1]$ (resp. $alpha>|mathbb{E}[X_1]|$) the set $mathcal{H}_alpha$ (resp. $mathcal{Z}_alpha$) is non-empty, closed, unbounded above and below, stationary, and regenerative. The collections $(mathcal{H}_{alpha})_{alpha > mathbb{E}[X_1]}$ and $(mathcal{Z}_{alpha})_{alpha > |mathbb{E}[X_1]|}$ are increasing in $alpha$ and the regeneration property is compatible with these inclusions in that each family is a continuum of embedded regenerative sets in the sense of Bertoin. We show that $(sup{t < 0 : t in mathcal{H}_alpha})_{alpha > mathbb{E}[X_1]}$ is a c`adl`ag, nondecreasing, pure jump process with independent increments and determine the intensity measure of the associated Poisson process of jumps. We obtain a similar result for $(sup{t < 0 : t in mathcal{Z}_alpha})_{alpha > |beta|}$ when $(X_t)_{t in mathbb{R}}$ is a (two-sided) Brownian motion with drift $beta$.
We review the theory of regenerative processes, which are processes that can be intuitively seen as comprising of i.i.d. cycles. Although we focus on the classical definition, we present a more general definition that allows for some form of dependen
A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the L{e}vy measure
We consider the discrete Gaussian Free Field (DGFF) in scaled-up (square-lattic
For $widetilde{cal R} = 1 - exp(- {cal R})$ a random closed set obtained by exponential transformation of the closed range ${cal R}$ of a subordinator, a regenerative composition of generic positive integer $n$ is defined by recording the sizes of cl
We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for t