arXiv subjects

Zakaria Derbazi

Publications and source records attributed to Zakaria Derbazi.

9 recordsLinked to original sources

arXiv2026

A Probabilistic Sign Rule for Quotients of Positive Series and Integral Transforms

This paper develops a probabilistic sign rule for quotients of functions represented by positive series or integrals. For a function in this class, normalising the summand function in the series case or the integrand function in the integral case induces a probability law under which parameter log-derivatives of the function are expressed as moments of kernels, the log-derivatives of the same summand or integrand function with respect to the same parameters. The resulting moment identities reduce quotient monotonicity, log-supermodularity, and log-convexity to sign criteria based on kernel monotonicity, stochastic ordering of the induced laws, and covariance or variance identities. The criteria are applied to generalised hypergeometric, Stieltjes-transform, and Prabhakar quotients, yielding new Turán inequalities, two-sided Stieltjes bounds, and a local failure threshold for a monotonicity conjecture for the zero-balanced Gauss function.

math.CA↗

arXiv2026

Kernel Characterisations of Stochastic Orders Within Parametric Density Families

We develop kernel criteria for the likelihood-ratio, hazard-rate, usual stochastic, and relative log-concavity orders in parametric families of univariate probability laws with densities. The score is the derivative of the log density with respect to the parameter, and a kernel equals the score up to an additive term depending only on the parameter. Kernel monotonicity gives likelihood-ratio order, kernel concavity gives relative log-concavity, and two tail-conditional mean inequalities give the hazard-rate and usual stochastic orders. The same construction applies along joint-parameter paths and to comparisons between two laws whose densities admit parameter-dependent factors, where the log-factor ratio is used as the kernel. For compound sums with a random number of i.i.d. terms, the induced kernel is the posterior mean of the kernel of the summand count. The applications recover standard one-parameter orderings, give likelihood-ratio comparisons for compound laws, and handle nonmonotone examples through the tail-conditional criteria.

math.PR↗

arXiv2025

On a connection between total positivity and Bernoulli stopping problems

Consider a discrete-time optimal selection problem where one observes a sequence of independent Bernoulli trials and receives a nonnegative reward upon stopping on a success. The aim is to find a single-choice strategy that maximises the expected payoff. These Bernoulli stopping problems are characterised by two key properties: (i) a recurrence relation connecting the reward sequence to the continuation payoff sequence, and (ii) the total positivity of the Markov chain embedded in success epochs of the trials. The recurrence is fundamental in proving the optimality of the myopic strategy under unimodal continuation payoff sequence, while the total positivity ensures that the expectation of a quasi-unimodal function of the chain remains quasi-unimodal with respect to the initial state. In particular, if the number of successes is finite almost surely, the quasi-unimodality of the reward sequence is sufficient for the myopic rule to be optimal. Illustrative examples are given in various last-success settings.

math.PR↗

arXiv2025

On Preserving or Reversing Higher-Order Unimodality and Convexity by Sign-Regular Kernels

This work investigates preserving and reversing unimodality and convexity properties for sequences under transformations defined by sign-regular kernels. It is shown that these transformations only preserve these properties if the kernels are totally positive of order three or their additive inverse is totally negative of order three. In contrast, these transformations reverse these properties if the underlying kernel is totally negative or if its additive inverse is a totally positive kernel, both of order three. Furthermore, these results are extended to higher-order convex and multimodal sequences. These findings, which expand upon Karlin's earlier results on convexity, form the basis for deriving sufficient conditions for the preservation or reversal of higher-order convexity or generalised unimodality of a quotient of sequences, where both the numerator and denominator are transformations by the same sign-regular kernel.

math.CA↗

arXiv2024

Sum the Probabilities to $m$ and Stop

This work investigates the optimal selection of the $m$th last success in a sequence of $n$ independent Bernoulli trials. We propose a threshold strategy that is $\varepsilon$-optimal under minimal assumptions about the monotonicity of the trials' success probabilities. This new strategy ensures stopping at most one step earlier than the optimal rule. Specifically, the new threshold coincides with the point where the sum of success probabilities in the remaining trials equals $m$. We show that the underperformance of the new rule, in comparison to the optimal one, is of the order $O(n^{-2})$ in the case of a Karamata-Stirling success profile with parameter $θ> 0$ where $p_k = θ/ (θ+ k - 1)$ for the $k$th trial. We further leverage the classical weak convergence of the number of successes in the trials to a Poisson random variable to derive the asymptotic solution of the stopping problem. Finally, we present illustrative examples highlighting the close performance between the two rules.

math.PR↗

arXiv2024

An Optimal Selection Problem Associated with the Poisson Process

Cowan and Zabczyk (1978) introduced a continuous-time generalisation of the secretary problem, where offers arrive at epochs of a homogeneous Poisson process. We expand their work to encompass the last-success problem under the Karamata-Stirling success profile. In this setting, the $k$th trial is a success with probability $p_k=θ/(θ+k-1)$, where $θ> 0$. In the best-choice setting ($θ=1$), the myopic strategy is optimal, and the proof hinges on verifying the monotonicity of certain critical roots. We extend this crucial result to the last-success case by exploiting a connection to the sign of the derivative in the first parameter of a quotient of Kummer's hypergeometric functions. Additionally, we establish an Edmundson-Madansky inequality applicable to Poisson random variables. This result enables us to adopt a probabilistic approach to derive bounds and asymptotics of the critical roots. This strengthens and improves the findings of Ciesielski and Zabczyk (1979).

math.PR↗

arXiv2023

Records in the Infinite Occupancy Scheme

We consider the classic infinite occupancy scheme, where balls are thrown in boxes independently, with probability $p_j$ of hitting box $j$. Each time a box receives its first ball we speak of a record and, more generally, call an $r$-record every event when a box receives its $r$th ball. Assuming that the sequence $(p_j)$ is not decaying too fast, we show that after many balls have been thrown, the suitably scaled point process of $r$-record times is approximately Poisson. The joint convergence of $r$-record processes is argued under a condition of regular variation.

math.PR↗

arXiv2022

The Last-Success Stopping Problem with Random Observation Times

Suppose $N$ independent Bernoulli trials are observed sequentially at random times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. We focus on the version of the problem where the $k^\text{th}$ trial is a success with probability $p_k=θ/(θ+k-1)$ and the prior distribution of $N$ is negative binomial with shape parameter $ν$. Exploring properties of the Gaussian hypergeometric function, we find that the myopic stopping strategy is optimal if and only if $ν\geqθ$. We derive formulas to assess the winning probability and discuss limit forms of the problem for large $N$.

math.PR↗

arXiv2021

Trapping the Ultimate Success

We introduce a betting game, where the gambler aims to guess the last success epoch from past observed data. The player may bet on the event that no further successes occur, or choose a `trap' which is any span of future times. In the latter case winning is achieved if the last success turns out to be the only one falling in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success in a finite sequence of trials. We use this connection to analyse the problem of stopping at the last record for trials paced by a Polya-Lundberg process with log-series distribution of the total number of trials.

math.PR↗