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Philip A. Ernst

Publications and source records attributed to Philip A. Ernst.

15 recordsLinked to original sources
arXiv2026

Asymptotic optimality of dynamic first-fit packing on the half-axis

We revisit a classical problem in dynamic storage allocation. Items arrive in a linear storage medium, modeled as a half-axis, at a Poisson rate $r$ and depart after an independent exponentially distributed unit mean service time. The arriving item sizes (lengths) are assumed to be independent and identically distributed (i.i.d.) from a common distribution $H$. A widely employed algorithm for allocating the items is the "first-fit" discipline, namely, each arriving item is placed in the left-most vacant interval large enough to accommodate it. In a seminal 1985 paper, Coffman, Kadota, and Shepp ([6]) proved that in the special case of unit length items (i.e. degenerate $H$), as $r$ tends towards infinity, the first-fit algorithm is asymptotically optimal in the following sense: the steady-state ratio of expected "empty space" (gaps between items) to expected occupied space tends towards $0$. In a sequel to [6], Coffman, Kadota, and Shepp ([5]) conjectured that the first-fit discipline is also asymptotically optimal for non-degenerate $H$. In this paper we provide the first proof of first-fit asymptotic optimality for non-degenerate distributions $H$ of item sizes. Our main result is for the case when $H$ is concentrated on countably many positive real sizes forming an increasing sequence that is either finite or goes to infinity, with the average item size being finite. We prove that under the first-fit discipline, as $r$ tends towards infinity, the steady-state packing configuration (scaled down by $r$) converges in distribution to the limiting packing configuration with smaller items on the left, larger items on the right, and with no gaps between. In particular, this proves asymptotic optimality of first-fit in the following sense: if $P$ is the expected occupied space, then in steady-state the empty space (scaled down by $r$) in $[0,P]$ vanishes.

math.PR↗
arXiv2025

Exact and asymptotic distribution theory for the empirical correlation of two AR(1) processes with Gaussian increments

This paper begins with a study of the exact distribution of the empirical correlation of two independent AR(1) processes with Gaussian increments. We proceed to develop rates of convergence for the distribution of the scaled empirical correlation to the standard Gaussian distribution in both Wasserstein distance and Kolmogorov distance. Given $n$ data points, we prove the convergence rate in Wasserstein distance is $n^{-1/2}$ and the convergence rate in Kolmogorov distance is $n^{-1/2} \sqrt{\ln n}$. We conclude by extending these results to two AR(1) processes with correlated Gaussian increments.

math.ST↗
arXiv2025

The value of partial information

We investigate a pricing rule that is applicable for streams of income or contingent claim liabilities and study how this rule changes under additional insider-type information that an investor might obtain. Considering a model where the risky asset might have jumps, we obtain an explicit form of the associated state price density for the three different types of agents considered in [ER20]: one who has no information about the jumps, one who knows in advance exactly when the each jump will occur, and one who has no information about the size of the jumps but has partial information about the size of each jump. For each of these agents, we provide characterizations of the pricing rule and establish a representation formula, allowing us to quantify the value of partial information for streams of labor income or contingent claim liabilities. Our work is motivated by finding and characterizing a pricing rule that, both with or without partial information about jumps, assigns different values of information for different income streams or contingent claim liabilities.

q-fin.MF↗
arXiv2024

Exact expressions for the maximal probability that all $k$-wise independent bits are 1

Let $M(n, k, p)$ denote the maximum probability of the event $X_1 = X_2 = \cdots = X_n=1$ under a $k$-wise independent distribution whose marginals are Bernoulli random variables with mean $p$. A long-standing question is to calculate $M(n, k, p)$ for all values of $n,k,p$. This question has been partially addressed by several authors, primarily with the goal of answering asymptotic questions. The present paper focuses on obtaining exact expressions for this probability. To this end, we provide closed-form formulas of $M(n,k,p)$ for $p$ near 0 as well as $p$ near 1.

math.PR↗
arXiv2024

The Gapeev-Shiryaev Conjecture

The Gapeev-Shiryaev conjecture (originating in Gapeev and Shiryaev (2011) and Gapeev and Shiryaev (2013)) can be broadly stated as follows: Monotonicity of the signal-to-noise ratio implies monotonicity of the optimal stopping boundaries. The conjecture was originally formulated both within (i) sequential testing problems for diffusion processes (where one needs to decide which of the two drifts is being indirectly observed) and (ii) quickest detection problems for diffusion processes (where one needs to detect when the initial drift changes to a new drift). In this paper we present proofs of the Gapeev-Shiryaev conjecture both in (i) the sequential testing setting (under Lipschitz/Holder coefficients of the underlying SDEs) and (ii) the quickest detection setting (under analytic coefficients of the underlying SDEs). The method of proof in the sequential testing setting relies upon a stochastic time change and pathwise comparison arguments. Both arguments break down in the quickest detection setting and get replaced by arguments arising from a stochastic maximum principle for hypoelliptic equations (satisfying Hormander's condition) that is of independent interest. Verification of the Gapeev-Shiryaev conjecture establishes the fact that sequential testing and quickest detection problems with monotone signal-to-noise ratios are amenable to known methods of solution.

math.PR↗
arXiv2023

Quickest Real-Time Detection of Multiple Brownian Drifts

Consider the motion of a Brownian particle in $n$ dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, exactly $k$ of the coordinate processes get a (known) non-zero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which the $k$ coordinate processes get the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion without drift. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection. The elliptic case $k=1$ has been settled in Ernst and Peskir (2022) where the hypoelliptic case $1 < k < n$ resolved in the present paper was left open (the case $k = n$ reduces to the classic case $n=1$ having a known solution). We also show that the methodology developed solves the problem in the general case where exactly $k$ is relaxed to any number of the coordinate processes getting the drift. To our knowledge this is the first time that such a multi-dimensional hypoelliptic problem has been solved exactly in the literature.

math.PR↗
arXiv2023

Yule's "nonsense correlation" solved: Part II

In 1926, G. Udny Yule considered the following: given a sequence of pairs of random variables $\{X_k,Y_k \}$ ($k=1,2, \ldots, n$), and letting $X_i = S_i$ and $Y_ i= S'_i$ where $S_i$ and $S'_i$ are the partial sums of two independent random walks, what is the distribution of the empirical correlation coefficient \begin{equation*} ρ_n = \frac{\sum_{i=1}^n S_i S^\prime_i - \frac{1}{n}(\sum_{i=1}^n S_i)(\sum_{i=1}^n S^\prime_i)}{\sqrt{\sum_{i=1}^n S^2_i - \frac{1}{n}(\sum_{i=1}^n S_i)^2}\sqrt{\sum_{i=1}^n (S^\prime_i)^2 - \frac{1}{n}(\sum_{i=1}^n S^\prime_i)^2}}? \end{equation*} Yule empirically observed the distribution of this statistic to be heavily dispersed and frequently large in absolute value, leading him to call it "nonsense correlation." This unexpected finding led to his formulation of two concrete questions, each of which would remain open for more than ninety years: (i) Find (analytically) the variance of $ρ_n$ as $n \rightarrow \infty$ and (ii): Find (analytically) the higher order moments and the density of $ρ_n$ as $n \rightarrow \infty$. In 2017, Ernst, Shepp, and Wyner considered the empirical correlation coefficient \begin{equation*} ρ:= \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1 W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - (\int_0^1W_1(t) dt)^2} \sqrt{\int_0^1 W^2_2(t) dt - (\int_0^1W_2(t) dt)^2}}\end{equation*} of two independent Wiener processes $W_1,W_2$, the limit to which $ρ_n$ converges weakly, as was first shown by Phillips (1986). Using tools from integral equation theory, Ernst et al. (2017) closed question (i) by explicitly calculating the second moment of $ρ$ to be .240522. This paper begins where Ernst et al. (2017) leaves off. We succeed in closing question (ii) by explicitly calculating all moments of $ρ$ (up to order 16).

math.ST↗
arXiv2021

Yule's "nonsense correlation" for Gaussian random walks

The purpose of this paper is to provide an exact formula for the second moment of the empirical correlation of two independent Gaussian random walks as well as implicit formulas for higher moments. The proofs are based on a symbolically tractable integro-differential representation formula for the moments of any order in a class of empirical correlations, first established by Ernst et al. (2019) and investigated previously in Ernst et al. (2017). We also provide rates of convergence of the empirical correlation of two independent Gaussian random walks to the empirical correlation of two independent Wiener processes, by exploiting the explicit nature of the computations used for the moments. At the level of distributions, in Wasserstein distance, the convergence rate is the inverse $n^{-1}$ of the number of data points $n$. This holds because we represent and couple the discrete and continuous correlations on a common probability space, where we establish convergence in $L^1$ at the rate $n^{-1}$.

math.PR↗
arXiv2021

On the diameter of the stopped spider process

We consider the Brownian ``spider process'', also known as Walsh Brownian motion, first introduced in the epilogue of Walsh 1978. The paper provides the best constant $C_n$ for the inequality $$ E D_τ\leq C_n \sqrt{E τ},$$ where $τ$ is the class of all adapted and integrable stopping times and $D$ denotes the diameter of the spider process measured in terms of the British rail metric. The proof relies on the explicit identification of the value function for the associated optimal stopping problem.

math.PR↗
arXiv2021

Fiscal stimulus as an optimal control problem

During the Great Recession, Democrats in the United States argued that government spending could be utilized to "grease the wheels" of the economy in order to create wealth and to increase employment; Republicans, on the other hand, contended that government spending is wasteful and discouraged investment, thereby increasing unemployment. Today, in 2020, we find ourselves in the midst of another crisis where government spending and fiscal stimulus is again being considered as a solution. In the present paper, we address this question by formulating an optimal control problem generalizing the model of Radner & Shepp (1996). The model allows for the company to borrow continuously from the government. We prove that there exists an optimal strategy; rigorous verification proofs for its optimality are provided. We proceed to prove that government loans increase the expected net value of a company. We also examine the consequences of different profit-taking behaviors among firms who receive fiscal stimulus.

econ.GN↗
arXiv2021

The least favorable noise

Suppose that a random variable $X$ of interest is observed perturbed by independent additive noise $Y$. This paper concerns the "the least favorable perturbation" $\hat Y_\ep$, which maximizes the prediction error $E(X-E(X|X+Y))^2$ in the class of $Y$ with $ \var (Y)\leq \ep$. We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where $X$ is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier $Y$ makes prediction worse.

math.PR↗
arXiv2020

Quickest Real-Time Detection of a Brownian Coordinate Drift

Consider the motion of a Brownian particle in two or more dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, one of the coordinate processes gets a (known) non-zero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which a coordinate process gets the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the initial motion without drift. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection. To our knowledge this is the first time that such a problem has been solved exactly in the literature.

math.PR↗
arXiv2020

The rencontre problem

Let $\left\{X^{1}_k\right\}_{k=1}^{\infty}, \left\{X^{2}_k\right\}_{k=1}^{\infty}, \cdots, \left\{X^{d}_k\right\}_{k=1}^{\infty}$ be $d$ independent sequences of Bernoulli random variables with success-parameters $p_1, p_2, \cdots, p_d$ respectively, where $d \geq 2$ is a positive integer, and $ 0<p_j<1$ for all $j=1,2,\cdots,d.$ Let \begin{equation*} S^{j}(n) = \sum_{i=1}^{n} X^{j}_{i} = X^{j}_{1} + X^{j}_{2} + \cdots + X^{j}_{n}, \quad n =1,2 , \cdots. \end{equation*} We declare a "rencontre" at time $n$, or, equivalently, say that $n$ is a "rencontre-time," if \begin{equation*} S^{1}(n) = S^{2}(n) = \cdots = S^{d}(n). \end{equation*} We motivate and study the distribution of the first (provided it is finite) rencontre time.

math.PR↗
arXiv2018

Thick distribution tails in models of cancer secondary tumors

Recent progress in microdissection and in DNA sequencing has enabled subsampling of multi-focal cancers in organs such as the liver in several hundred spots, helping to determine the pattern of mutations in each of these spots. This has led to the construction of genealogies of the primary, secondary, tertiary and so forth, foci of the tumor. These studies have led to diverse conclusions concerning the Darwinian (selective) or neutral evolution in cancer. Mathematical models of development of multifocal tumors have been developed to support these claims. We report a model of development of a multifocal tumor, which is a mathematically rigorous refinement of a model of Ling et al. (2015). Guided by numerical studies and simulations, we show that the rigorous model, in the form of an infinite-type branching process, displays distributions of tumors size which have heavy tails and moments that become infinite in finite time. To demonstrate these points, we obtain bounds on the tails of the distributions of the process and infinite-series expression for the first moments. In addition to its inherent mathematical interest, the model is corroborated by recent reports of apparent super-exponential growth in cancer metastases.

q-bio.PE↗