A central limit theorem for temporally non-homogenous Markov chains with applications to dynamic programming

We prove a central limit theorem for a class of additive processes that arise naturally in the theory of finite horizon Markov decision problems. The main theorem generalizes a classic result of Dobrushin (1956) for temporally non-homogeneous Markov chains, and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain. We show through several examples that this added flexibility gives one a direct path to asymptotic normality of the optimal total reward of finite horizon Markov decision problems. The same examples also explain why such results are not easily obtained by alternative Markovian techniques such as enlargement of the state space.Comment: 27 pages, 1 figur

Quickest Online Selection of an Increasing Subsequence of Specified Size

Given a sequence of independent random variables with a common continuous distribution, we consider the online decision problem where one seeks to minimize the expected value of the time that is needed to complete the selection of a monotone increasing subsequence of a prespecified length $n$. This problem is dual to some online decision problems that have been considered earlier, and this dual problem has some notable advantages. In particular, the recursions and equations of optimality lead with relative ease to asymptotic formulas for mean and variance of the minimal selection time.Comment: 17 page

Optimal Online Selection of a Monotone Subsequence: a Central Limit Theorem

Consider a sequence of $n$ independent random variables with a common continuous distribution $F$, and consider the task of choosing an increasing subsequence where the observations are revealed sequentially and where an observation must be accepted or rejected when it is first revealed. There is a unique selection policy $\pi_n^*$ that is optimal in the sense that it maximizes the expected value of $L_n(\pi_n^*)$, the number of selected observations. We investigate the distribution of $L_n(\pi_n^*)$; in particular, we obtain a central limit theorem for $L_n(\pi_n^*)$ and a detailed understanding of its mean and variance for large $n$. Our results and methods are complementary to the work of Bruss and Delbaen (2004) where an analogous central limit theorem is found for monotone increasing selections from a finite sequence with cardinality $N$ where $N$ is a Poisson random variable that is independent of the sequence.Comment: 26 page

Optimal Online Selection of an Alternating Subsequence: A Central Limit Theorem

We analyze the optimal policy for the sequential selection of an alternating subsequence from a sequence of n independent observations from a continuous distribution F, and we prove a central limit theorem for the number of selections made by that policy. The proof exploits the backward recursion of dynamic programming and assembles a detailed understanding of the associated value functions and selection rules

Beardwood-Halton-Hammersly Theorem for Stationary Ergodic Sequences: A Counterexample

We construct a stationary ergodic process X1,X2,…such that each Xt has the uniform distribution on the unit square and the length Ln of the shortest path through the points X1,X2,…,Xn is not asymptotic to a constant times the square root of n. In other words, we show that the Beardwood, Halton, and Hammersley theorem does not extend from the case of independent uniformly distributed random variables to the case of stationary ergodic sequences with uniform marginal distributions

Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence

We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences

Online Selection of Alternating Subsequences from a Random Sample

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence

Formation et esprit d'entreprendre chez les étudiants

Cette contribution s'attache à étudier entre esprit d'entreprendre et domaine d'études chez les étudiants français. L'esprit d'entreprendre est le substrat dans lequel peut germer l'intention qui pourra donner lieu à une création d'entreprise. Nous choisissons de l'appréhender à partir d'une grille de lecture dérivée du modèle psychosocial du comportement planifié d'Ajzen, enrichie par des variables reflétant l'implication dans la vie associative et les opinions sur les formations à la création d'entreprise.esprit d'entreprendre ; domaine d'étude ; étudiants français

Humoral Responses against BQ.1.1 Elicited after Breakthrough Infection and SARS-CoV-2 mRNA Vaccination.

The Omicron BQ.1.1 variant is now the major SARS-CoV-2 circulating strain in many countries. Because of the many mutations present in its Spike glycoprotein, this variant is resistant to humoral responses elicited by monovalent mRNA vaccines. With the goal to improve immune responses against Omicron subvariants, bivalent mRNA vaccines have recently been approved in several countries. In this study, we measure the capacity of plasma from vaccinated individuals, before and after a fourth dose of mono- or bivalent mRNA vaccine, to recognize and neutralize the ancestral (D614G) and the BQ.1.1 Spikes. Before and after the fourth dose, we observe a significantly better recognition and neutralization of the ancestral Spike. We also observe that fourth-dose vaccinated individuals who have been recently infected better recognize and neutralize the BQ.1.1 Spike, independently of the mRNA vaccine used, than donors who have never been infected or have an older infection. Our study supports that hybrid immunity, generated by vaccination and a recent infection, induces higher humoral responses than vaccination alone, independently of the mRNA vaccine used