On binomial thinning and mixing
In this paper we consider the notions of binomial thinning, binomial mixing, their generalizations, certain interplay between them, associated limit theorems and provide various examples.
math.PR↗
arXiv subjects
Andreas Löpker
Publications and source records attributed to Andreas Löpker.
A Multiplicative Version of the Lindley Recursion
This paper presents an analysis of the stochastic recursion $W_{i+1} = [V_iW_i+Y_i]^+$ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability condition. Writing $Y_i=B_i-A_i$, for independent sequences of non-negative i.i.d.\ random variables $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$, and assuming $\{V_i\}_{i\in N_0}$ is an i.i.d. sequence as well (independent of $\{A_i\}_{i\in N_0}$ and $\{B_i\}_{i\in N_0}$), we then consider three special cases: (i) $V_i$ attains negative values only and $B_i$ has a rational LST, (ii) $V_i$ equals a positive value $a$ with certain probability $p\in (0,1)$ and is negative otherwise, and both $A_i$ and $B_i$ have a rational LST, (iii) $V_i$ is uniformly distributed on $[0,1]$, and $A_i$ is exponentially distributed. In all three cases we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.
math.PR↗
Weak Convergence of Subordinators to Extremal Processes
For certain subordinators $(X_t)_{t\ge 0}$ it is shown that the process $(-t\log X_{ts})_{s>0}$ tends to an extremal process $(\hatη_s)_{s>0}$ in the sense of convergence of the finite dimensional distributions. Additionally it is also shown that $(z\wedge(-t\log X_{ts}))_{s\ge 0}$ converges weakly to $(z\wedge\hatη_s)_{s\ge0}$ in $\mathcal{D}[0,\infty)$, the space of càdlàg functions equipped with Skorohod's $J_1$ metric.
math.PR↗
On the small-time behavior of subordinators
We prove several results on the behavior near t=0 of $Y_t^{-t}$ for certain $(0,\infty)$-valued stochastic processes $(Y_t)_{t>0}$. In particular, we show for Lévy subordinators that the Pareto law on $[1,\infty)$ is the only possible weak limit and provide necessary and sufficient conditions for the convergence. More generally, we also consider the weak convergence of $tL(Y_t)$ as $t\to0$ for a decreasing function $L$ that is slowly varying at zero. Various examples demonstrating the applicability of the results are presented.
math.ST↗
On Time Reversal of Piecewise Deterministic Markov Processes
We study the time reversal of a general PDMP. The time reversed process is defined as $X_{(T-t)-}$, where $T$ is some given time and $X_t$ is a stationary PDMP. We obtain the parameters of the reversed process, like the jump intensity and the jump measure.
math.PR↗