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Ira W. Herbst

Publications and source records attributed to Ira W. Herbst.

6 recordsLinked to original sources
arXiv2026

Spectral and dynamical results related to certain non-integer base expansions on the unit interval

We consider certain non-integer base $β$-expansions of Parry's type and we study various properties of the transfer (Perron-Frobenius) operator $\mathcal{P}:L^p([0,1])\mapsto L^p([0,1])$ with $p\geq 1$ and its associated composition (Koopman) operator, which are induced by a discrete dynamical system on the unit interval related to these $β$-expansions. We show that if $f$ is Lipschitz, then the iterated sequence $\{\mathcal{P}^N f\}_{N\geq 1}$ converges exponentially fast (in the $L^1$ norm) to an invariant state corresponding to the eigenvalue $1$ of $\mathcal{P}$. This "attracting" eigenvalue is not isolated: for $1\leq p\leq 2$ we show that the point spectrum of $\mathcal{P}$ also contains the whole open complex unit disk and we explicitly construct some corresponding eigenfunctions.

math.SP↗
arXiv2024

The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let $X=\sum_{k=1}^\infty X_k β^{-k}$ be the base-$β$ expansion of a continuous random variable $X$ on the unit interval where $β$ is the positive solution to $β^n = 1 + β+ \cdots + β^{n-1}$ for an integer $n\ge 2$ (i.e., $β$ is a generalization of the golden mean for which $n=2$). We study the asymptotic distribution and convergence rate of the scaled remainder $\sum_{k=1}^\infty X_{m+k} β^{-k}$ when $m$ tends to infinity.

math.PR↗
arXiv2023

How many digits are needed?

Let $X_1,X_2,...$ be the digits in the base-$q$ expansion of a random variable $X$ defined on $[0,1)$ where $q\ge2$ is an integer. For $n=1,2,...$, we study the probability distribution $P_n$ of the (scaled) remainder $T^n(X)=\sum_{k=n+1}^\infty X_k q^{n-k}$: If $X$ has an absolutely continuous CDF then $P_n$ converges in the total variation metric to the Lebesgue measure $μ$ on the unit interval. Under weak smoothness conditions we establish first a coupling between $X$ and a non-negative integer valued random variable $N$ so that $T^N(X)$ follows $μ$ and is independent of $(X_1,...,X_N)$, and second exponentially fast convergence of $P_n$ and its PDF $f_n$. We discuss how many digits are needed and show examples of our results. The convergence results are extended to the case of a multivariate random variable defined on a unit cube.

math.PR↗
arXiv2022

Singular distribution functions for random variables with stationary digits

Let $F$ be the cumulative distribution function (CDF) of the base-$q$ expansion $\sum_{n=1}^\infty X_n q^{-n}$, where $q\ge2$ is an integer and $\{X_n\}_{n\geq 1}$ is a stationary stochastic process with state space $\{0,\ldots,q-1\}$. In a previous paper we characterized the absolutely continuous and the discrete components of $F$. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: $F$ is then either a uniform or a singular CDF on $[0,1]$. Moreover, we study mixtures of such models. In most cases expressions and plots of $F$ are given.

math.PR↗
arXiv2022

Characterization of random variables with stationary digits

Let $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$, and $F$ the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$. We show that stationarity of $\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by $F$ and use this to characterize the characteristic function of $X$ and the structure of $F$ in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of $F$ can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that $\mathrm{d} F$ is a Rajchman measure if and only if $F $ is the uniform CDF on $[0,1]$.

math.PR↗