arXiv subjects

James Allen Fill

Publications and source records attributed to James Allen Fill.

At least 19 recordsLinked to original sources

arXiv2026

Positive autocorrelation at unit lag for stationary random walk Metropolis-Hastings in ${\mathbb R}^d$

It is often asserted in the literature that one should expect positive autocorrelation for random walk Metropolis-Hastings (RWMH), especially if the typical proposal step-size is small relative to the variability in the target density. In this paper, we consider a stationary RWMH chain ${\bf X}$ taking values in $d$-dimensional Euclidean space and (subject only to the existence of densities with respect to Lebesgue measure) with general target distribution having finite second moment and general proposal random walk step-distribution. We prove, for any nonzero vector ${\bf c}$, strict positivity of the autocorrelation function at unit lag for the stochastic process $\langle{\bf c},{\bf X}\rangle$, that is, \[{\operatorname{Corr}}(\langle{\bf c},{\bf X}_0\rangle,\langle{\bf c},{\bf X}_1\rangle)>0,\] and we establish the same result, but with weak inequality (which can in some cases be equality) when the state space for ${\bf X}$ is changed to the integer grid ${\mathbb Z}^d$. Further, for ${\bf c}\neq{\bf 0}$ we establish the sharp lower bound \[{\operatorname{Corr}}(\langle{\bf c},{\bf X}_0\rangle,\langle{\bf c},{\bf X}_1\rangle)>\tfrac19\] on autocorrelation when we assume both that (i) the target density $\pi$ is spherically symmetric and unimodal in the specific sense that $\pi({\bf x})=\hat{\pi}(\|{\bf x}\|)$ for some nonincreasing function $\hat{\pi}$ on $[0,\infty)$ and that (ii) the proposal step-density is symmetric about ${\bf 0}$. We study the autocorrelation indirectly, by considering the incremental variance function (or incremental second-moment function) at unit lag. The same approach allows us also for $r\in[2,\infty)$ to upper-bound the incremental $r$th-absolute-moment function at unit lag. We give also closely related inequalities for the total variation distance between two distributions on ${\mathbb R}^d$ differing only by a location shift.

math.PR↗

arXiv2026

A new fine-scale Berry-Esseen-type Gumbel-limit theorem for multivariate maxima

For $d \geq 2$ and i.i.d. $d$-dimensional observations $\mathbf{X}^{(1)}, \mathbf{X}^{(2)}, \ldots$ with independent Exponential$(1)$ coordinates, let $\varphi_n$ denote the minimum $\ell^1$-norm among the maxima of $\{\mathbf{X}^{(1)}, \ldots, \mathbf{X}^{(n)}\}$. (A _maximum_ from this set is an observation $\mathbf{X}^{(k)}$ with $1 \leq k \leq n$ such that $\mathbf{X}^{(k)} \not\prec \mathbf{X}^{(i)}$ for all $1 \leq i \leq n$, where $\mathbf{x} \prec \mathbf{y}$ means that $x_j < y_j$ for $1 \leq j \leq d$.) Key roles in the study of multivariate Pareto records are played by $\varphi_n$ and by the more easily handled maximum with the maximum $\ell^1$-norm. Fill, Naiman, and Sun (2024) proved that \[ \varphi_n = \ln n - \ln \ln \ln n - \ln(d - 1) + O_{\mathrm{p}}\!\left( \frac{1}{\ln \ln n} \right), \] where $Z_n = O_{\mathrm{p}}(a_n)$ means that $Z_n / a_n$ is bounded in probability, and conjectured that \[ (\ln \ln n) \left(\varphi_n - [\ln n - \ln \ln \ln n - \ln(d - 1)] \right) \] has a nondegenerate limiting distribution, suggesting that the limiting distribution might be that of $ - G$, where $G$ has a Gumbel distribution with location $ - \frac{\ln[(d - 1)!]}{d - 1}$ and scale $\frac{1}{d - 1}$. In the present paper we prove a Berry-Esseen-type theorem for this convergence in distribution, thereby establishing a very sharp result for $\varphi_n$.

math.PR↗

arXiv2025

An interesting spectral gap problem, from Jim Fill

At the request of Laszlo Babai, founder and an editor of the free online journal Theory of Computing (ToC), theoryofcomputing.org, in August, 2025, I am posting on the arXiv, essentially unedited and not updated, a combination of two closely related sets of unpublished notes from 2003. ToC is keen on publishing links to all bibliography items, and a paper soon to be published there makes progress on a conjecture in my 2003 notes. The sections "The problem", "Evidence in favor of the conjecture", "Facts about the spectral structure of the matrix $K$", and "Stronger conjectures" previously formed a document entitled "An interesting spectral gap problem, from Jim Fill". The sections "Introduction: Self-organizing lists" and "The move-ahead-$1$ (MA1) rule" formed a document entitled "Background on the gap problem". The two sets of notes have inspired some research, including (as one example, with no attempt here at a literature survey) a 2022 paper by Bhakta, Miracle, Randall, and Streib cited in this arXiv document.

math.PR↗

arXiv2025

Disintegration theorem for multifunctions, with applications to empirical Wasserstein distances and average-case statistical bounds

We prove a generalisation of the disintegration theorem to the setting of multifunctions between Polish probability spaces. Whereas the classical disintegration theorem guarantees the disintegration of a probability measure along the partition of the underlying space by the fibres of a measurable function, our theorem gives necessary and sufficient conditions for the measure to disintegrate along a cover of the underlying space defined by the fibres of a measurable multifunction. Building on this theorem, we introduce a new statistical notion: We declare a metric Polish probability space to be asymptotically disintegrable if $n$ i.i.d.-centred balls of decreasing radius carry a disintegration of the measure with probability tending to unity as $n\rightarrow\infty$. We give a number of both $1$-dimensional and higher-dimensional examples of asymptotically disintegrable spaces with associated quantitative rates, as well as a strong counterexample. Finally, we give two applications of the notion of asymptotic disintegrability. First, we prove that asymptotically disintegrable spaces admit an easy high-probability quantification of the law of large numbers in Wasserstein space, which in all dimensions either recovers or improves upon the best known rates in some regimes, and is never any worse than existing rates by more than a factor of 2 in the exponent of $n$, where $n$ is the number of sample points. Second, we prove that any asymptotically disintegrable space admits a high-probability bound on the error in approximating the expectation of any Lipschitz function by its empirical average over an i.i.d.\ sample. The bound is average-case in the sense that it depends only on the empirical average of the local Lipschitz constants of the function, rather than the global Lipschitz constant as obtained by Kantorovich-Rubinstein duality.

math.PR↗

arXiv2024

Convergence of the QuickVal Residual

QuickSelect (aka Find), introduced by Hoare (1961), is a randomized algorithm for selecting a specified order statistic from an input sequence of $n$ objects, or rather their identifying labels usually known as keys. The keys can be numeric or symbol strings, or indeed any labels drawn from a given linearly ordered set. We discuss various ways in which the cost of comparing two keys can be measured, and we can measure the efficiency of the algorithm by the total cost of such comparisons. We define and discuss a closely related algorithm known as QuickVal and a natural probabilistic model for the input to this algorithm; QuickVal searches (almost surely unsuccessfully) for a specified population quantile $\alpha \in [0, 1]$ in an input sample of size $n$. Call the total cost of comparisons for this algorithm $S_n$. We discuss a natural way to define the random variables $S_1, S_2, \ldots$ on a common probability space. For a general class of cost functions, Fill and Nakama (2013) proved under mild assumptions that the scaled cost $S_n / n$ of QuickVal converges in $L^p$ and almost surely to a limit random variable $S$. For a general cost function, we consider what we term the QuickVal residual: \[\rho_n := \frac{S_n}n - S.\] The residual is of natural interest, especially in light of the previous analogous work on the sorting algorithm QuickSort. In the case $\alpha = 0$ of QuickMin with unit cost per key-comparison, we are able to calculate -- \`a la Bindjeme and Fill (2012) for QuickSort -- the exact (and asymptotic) $L^2$-norm of the residual. We take the result as motivation for the scaling factor $\sqrt{n}$ for the QuickVal residual for general population quantiles and for general cost. We then prove in general (under mild conditions on the cost function) that $\sqrt{n}\,\rho_n$ converges in law to a scale-mixture of centered Gaussians, and we also prove convergence of moments.

math.PR↗

arXiv2024

On the probability of a Pareto record

Given a sequence of independent random vectors taking values in ${\mathbb R}^d$ and having common continuous distribution function $F$, say that the $n^{\rm \scriptsize th}$ observation sets a (Pareto) record if it is not dominated (in every coordinate) by any preceding observation. Let $p_n(F) \equiv p_{n, d}(F)$ denote the probability that the $n^{\rm \scriptsize th}$ observation sets a record. There are many interesting questions to address concerning $p_n$ and multivariate records more generally, but this short paper focuses on how $p_n$ varies with $F$, particularly if, under $F$, the coordinates exhibit negative dependence or positive dependence (rather than independence, a more-studied case). We introduce new notions of negative and positive dependence ideally suited for such a study, called negative record-setting probability dependence (NRPD) and positive record-setting probability dependence (PRPD), relate these notions to existing notions of dependence, and for fixed $d \geq 2$ and $n \geq 1$ prove that the image of the mapping $p_n$ on the domain of NRPD (respectively, PRPD) distributions is $[p^*_n, 1]$ (resp., $[n^{-1}, p^*_n]$), where $p^*_n$ is the record-setting probability for any continuous $F$ governing independent coordinates.

math.PR↗

arXiv2024

Sharpened localization of the trailing point of the Pareto record frontier

For $d\ge2$ and iid $d$-dimensional observations $X^{(1)},X^{(2)},\dots$ with independent Exponential$(1)$ coordinates, we revisit the study by Fill and Naiman (Electron. J. Probab., 2020) of the boundary (relative to the closed positive orthant), or "frontier", $F_n$ of the closed Pareto record-setting (RS) region \[ \mbox{RS}_n:=\{0\le x\in{\mathbb R}^d:x\not\prec X^{(i)}\mbox{\ for all $1\le i\le n$}\} \] at time $n$, where $0\le x$ means that $0\le x_j$ for $1\le j\le d$ and $x\prec y$ means that $x_j 0$ and $c_n\to\infty$ we have \[ {\mathbb P}(F_n^- -\ln n\in (-(2+\varepsilon)\ln\ln\ln n,c_n))\to 1 \] (describing typical behavior) and almost surely \[ \limsup \frac{F_n^- - \ln n}{\ln \ln n} \le 0 \quad \mbox{and} \quad \liminf \frac{F_n^- - \ln n}{\ln \ln \ln n} \in [-2, -1]. \] In this paper we use the theory of generators (minima of $F_n$) together with the first- and second-moment methods to improve considerably the trailing-point location results to \[ F_n^- - (\ln n - \ln \ln \ln n) \overset{\mathrm{P}}{\longrightarrow} - \ln(d - 1) \] (describing typical behavior) and, for $d \ge 3$, almost surely \begin{align*} &\limsup [F_n^- - (\ln n - \ln \ln \ln n)] \leq -\ln(d - 2) + \ln 2 \\ \mbox{and }&\liminf [F_n^- - (\ln n - \ln \ln \ln n)] \ge - \ln d - \ln 2. \end{align*}

math.PR↗

arXiv2022

Conditioned Galton-Watson trees: The shape functional, and more on the sum of powers of subtree sizes and its mean

For a complex number $\alpha$, we consider the sum of the $\alpha$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(\alpha)$ have been determined for $\Re\alpha \neq 0$, revealing a transition between a complex normal limiting distribution for $\Re\alpha < 0$ and a non-normal limiting distribution for $\Re\alpha > 0$. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case $\Re\alpha = 0$. The same results are also established in the case of the so-called shape functional $X_n'(0)$, which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. Additionally, we prove convergence of all moments in the case $\Re\alpha < 0$, where this result was previously missing, and establish new results about the asymptotic mean for real $\alpha < 1/2$. A novel feature for $\Re\alpha=0$ is that we find joint convergence for several $\alpha$ to independent limits, in contrast to the cases $\Re\alpha\neq0$, where the limit is known to be a continuous function of $\alpha$. Another difference from the case $\Re\alpha\neq0$ is that there is a logarithmic factor in the asymptotic variance when $\Re\alpha=0$; this holds also for the shape functional. The proofs are largely based on singularity analysis of generating functions.

math.PR↗

arXiv2021

Breaking Multivariate Records

For a sequence of i.i.d. $d$-dimensional random vectors with independent continuously distributed coordinates, say that the $n$th observation in the sequence sets a record if it is not dominated in every coordinate by an earlier observation; for $j \leq n$, say that the $j$th observation is a current record at time $n$ if it has not been dominated in every coordinate by any of the first $n$ observations; and say that the $n$th observation breaks $k$ records if it sets a record and there are $k$ observations that are current records at time $n - 1$ but not at time $n$. For general dimension $d$, we identify, with proof, the asymptotic conditional distribution of the number of (Pareto) records broken by an observation given that the observation sets a record. Fix $d$, and let ${\mathcal K}(d)$ be a random variable with this distribution. We show that the (right) tail of ${\mathcal K}(d)$ satisfies \[ {\mathbb P}({\mathcal K}(d) \geq k) \leq \exp\left[ - \Omega\!\left( k^{(d - 1) / (d^2 + d - 3)} \right) \right]\ \ \mbox{as $k \to \infty$} \] and \[ {\mathbb P}({\mathcal K}(d) \geq k) \geq \exp\left[ - O\!\left( k^{1 / (d - 1)} \right) \right]\ \ \mbox{as $k \to \infty$}. \] When $d = 2$, the description of ${\mathcal K}(2)$ in terms of a Poisson process agrees with the main result from Fill [Comb. Probab. Comput. 30 (2021) 105--123] that ${\mathcal K}(2)$ has the same distribution as ${\mathcal G} - 1$, where ${\mathcal G} \sim \mbox{Geometric$(1/2)$}$. Note that the lower bound on ${\mathbb P}({\mathcal K}(d) \geq k)$ implies that the distribution of ${\mathcal K}(d)$ is NOT (shifted) Geometric for any $d \geq 3$. We show that ${\mathbb P}({\mathcal K}(d) \geq 1) = \exp[-\Theta(d)]$ as $d \to \infty$; in particular, ${\mathcal K}(d) \to 0$ in probability as $d \to \infty$.

math.PR↗

arXiv2021

Density functions for QuickQuant and QuickVal

We prove that, for every $0 \leq t \leq 1$, the limiting distribution of the scale-normalized number of key comparisons used by the celebrated algorithm QuickQuant to find the $t$th quantile in a randomly ordered list has a Lipschitz continuous density function $f_t$ that is bounded above by $10$. Furthermore, this density $f_t(x)$ is positive for every $x > \min\{t, 1 - t\}$ and, uniformly in $t$, enjoys superexponential decay in the right tail. We also prove that the survival function $1 - F_t(x) = \int_x^{\infty}\!f_t(y)\,\mathrm{d}y$ and the density function $f_t(x)$ both have the right tail asymptotics $\exp [-x \ln x - x \ln \ln x + O(x)]$. We use the right-tail asymptotics to bound large deviations for the scale-normalized number of key comparisons used by QuickQuant.

math.PR↗

arXiv2021

The sum of powers of subtree sizes for conditioned Galton-Watson trees

We study the additive functional $X_n(α)$ on conditioned Galton-Watson trees given, for arbitrary complex $α$, by summing the $α$th power of all subtree sizes. Allowing complex $α$ is advantageous, even for the study of real $α$, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $\Reα< 0$, we prove that $X_n(α)$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in $α$, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence extensions of, limiting distribution results for $α$ in various regions of the complex plane. We focus mainly on the case where $\Reα> 0$, for which $X_n(α)$, suitably normalized, has a limiting distribution that is not normal but does not depend on the offspring distribution $ξ$ of the conditioned Galton-Watson tree, assuming only that $E[ξ] = 1$ and $0 < \mathrm{Var} [ξ] < \infty$. Under a weak extra moment assumption on $ξ$, we prove that the convergence extends to moments, ordinary and absolute and mixed, of all orders. At least when $\Reα> \frac12$, the limit random variable $Y(α)$ can be expressed as a function of a normalized Brownian excursion.

math.PR↗

arXiv2019

QuickSort: Improved right-tail asymptotics for the limiting distribution, and large deviations

We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $\log[1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(\log x)^2$; the corresponding order for the Janson (2015) bound is the lead order, $x \log x$. Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).

math.PR↗

arXiv2019

The Pareto Record Frontier

For iid $d$-dimensional observations $X^{(1)}, X^{(2)}, \ldots$ with independent Exponential$(1)$ coordinates, consider the boundary (relative to the closed positive orthant), or "frontier", $F_n$ of the closed Pareto record-setting (RS) region \[ \mbox{RS}_n := \{0 \leq x \in {\mathbb R}^d: x \not\prec X^{(i)}\ \mbox{for all $1 \leq i \leq n$}\} \] at time $n$, where $0 \leq x$ means that $0 \leq x_j$ for $1 \leq j \leq d$ and $x \prec y$ means that $x_j < y_j$ for $1 \leq j \leq d$. With $x_+ := \sum_{j = 1}^d x_j$, let \[ F_n^- := \min\{x_+: x \in F_n\} \quad \mbox{and} \quad F_n^+ := \max\{x_+: x \in F_n\}, \] and define the width of $F_n$ as \[ W_n := F_n^+ - F_n^-. \] We describe typical and almost sure behavior of the processes $F^+$, $F^-$, and $W$. In particular, we show that $F^+_n \sim \ln n \sim F^-_n$ almost surely and that $W_n / \ln \ln n$ converges in probability to $d - 1$; and for $d \geq 2$ we show that, almost surely, the set of limit points of the sequence $W_n / \ln \ln n$ is the interval $[d - 1, d]$. We also obtain modifications of our results that are important in connection with efficient simulation of Pareto records. Let $T_m$ denote the time that the $m$th record is set. We show that $F^+_{T_m} \sim (d! m)^{1/d} \sim F^-_{T_m}$ almost surely and that $W_{T_m} / \ln m$ converges in probability to $1 - d^{-1}$; and for $d \geq 2$ we show that, almost surely, the sequence $W_{T_m} / \ln m$ has $\liminf$ equal to $1 - d^{-1}$ and $\limsup$ equal to $1$.

math.PR↗

arXiv2019

Breaking Bivariate Records

We establish a fundamental property of bivariate Pareto records for independent observations uniformly distributed in the unit square. We prove that the asymptotic conditional distribution of the number of records broken by an observation given that the observation sets a record is Geometric with parameter 1/2.

math.PR↗

arXiv2019

Generating Pareto records

We present, (partially) analyze, and apply an efficient algorithm for the simulation of multivariate Pareto records. A key role is played by minima of the record-setting region (we call these generators) each time a new record is generated, and two highlights of our work are (i) efficient dynamic maintenance of the set of generators and (ii) asymptotic analysis of the expected number of generators at each time.

math.PR↗

arXiv2018

On the tails of the limiting QuickSort density

We give upper and lower asymptotic bounds for the left tail and for the right tail of the continuous limiting QuickSort density f that are nearly matching in each tail. The bounds strengthen results from a paper of Svante Janson (2015) concerning the corresponding distribution function F. Furthermore, we obtain similar bounds on absolute values of derivatives of f of each order.

math.PR↗

arXiv2017

A local limit theorem for Quicksort key comparisons via multi-round smoothing

As proved by Régnier and Rösler, the number of key comparisons required by the randomized sorting algorithm QuickSort to sort a list of $n$ distinct items (keys) satisfies a global distributional limit theorem. Fill and Janson proved results about the limiting distribution and the rate of convergence, and used these to prove a result part way towards a corresponding local limit theorem. In this paper we use a multi-round smoothing technique to prove the full local limit theorem.

math.PR↗

arXiv2015

Strong Stationary Duality for Diffusion Processes

We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simulations of our primal and dual diffusion processes. Lastly, we show how our new definition of diffusion duality allows the spectral theory of cutoff phenomena to extend naturally from birth-and-death Markov chains to the present diffusion context.

math.PR↗