arXiv subjects
Publications and source records attributed to Krishna B. Athreya.
We describe how certain properties of the extrema of the digits of Luroth expansions lead to a probabilistic proof of a limiting relation involving the Riemann zeta function and the Bernoulli triangles. We also discuss trimmed sums of Luroth digits. Our goal is to show how direct computations in this case lead to explicit formulas and some interesting discussions of special functions.
Let $\{X_n\}_{n\ge0}$ be a sequence of real valued random variables such that $X_n=ρ_n X_{n-1}+ε_n,~n=1,2,\ldots$, where $\{(ρ_n,ε_n)\}_{n\ge1}$ are i.i.d. and independent of initial value (possibly random) $X_0$. In this paper it is shown that, under some natural conditions on the distribution of $(ρ_1,ε_1)$, the sequence $\{X_n\}_{n\ge0}$ is regenerative in the sense that it could be broken up into i.i.d. components. Further, when $ρ_1$ and $ε_1$ are independent, we construct a non-parametric strongly consistent estimator of the characteristic functions of $ρ_1$ and $ε_1$.
Let $\{B(t), t \geq 0\}$ be a standard Brownian motion in $\mathbb{R}$. Let $T$ be the first return time to 0 after hitting 1, and $\{L(T,x), x \in \mathbb{R}\}$ be the local time process at time $T$ and level $x$. The distribution of $L(T,x)$ for each $x \in \mathbb{R}$ is determined. This is applied to the estimation of a $L^1$ integral on $\mathbb{R}$.
For a Radon measure $μ$ on $\bbR,$ we show that $L^{\infty}(μ)$ is invariant under the group of translation operators $T_t(f)(x) = {$f(x-t)$}\ (t \in \bbR)$ if and only if $μ$ is equivalent to Lebesgue measure $m$. We also give necessary and sufficient conditions for $L^p(μ),\1 \leq p < \infty,$ to be invariant under the group $\{T_t\}$ in terms of the Radon-Nikodym derivative w.r.t. $m$.
The two most important notions of fractal dimension are {\it Hausdorff dimension}, developed by Hausdorff (1919), and {\it packing dimension}, developed by Tricot (1982). Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of {\it gales}, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual of -- and every bit as simple as -- the gale characterization of Hausdorff dimension. Effectivizing our gale characterization of packing dimension produces a variety of {\it effective strong dimensions}, which are exact duals of the effective dimensions mentioned above. We develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression.