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Alexander Gnedin

Publications and source records attributed to Alexander Gnedin.

At least 37 records · Page 2Linked to original sources
arXiv2019

Asymptotics and Renewal Approximation in the Online Selection of Increasing Subsequence

We revisit the problem of maximising the expected length of increasing subsequence that can be selected from a marked Poisson process by an online strategy. Resorting to a natural size variable, the problem is represented in terms of a controlled partially deterministic Markov process with decreasing paths. Refining known estimates we obtain fairly complete asymptotic expansions for the moments, and using a renewal approximation give a novel proof of the central limit theorem for the length of selected subsequence under the optimal strategy.

math.PR↗
arXiv2018

On nested infinite occupancy scheme in random environment

We consider an infinite balls-in-boxes occupancy scheme with boxes organised in nested hierarchy, and random probabilities of boxes defined in terms of iterated fragmentation of a unit mass. We obtain a multivariate functional limit theorem for the cumulative occupancy counts as the number of balls approaches infinity. In the case of fragmentation driven by a homogeneous residual allocation model our result generalises the functional central limit theorem for the block counts in Ewens' and more general regenerative partitions.

math.PR↗
arXiv2017

The collision spectrum of $Λ$-coalescents

$Λ$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially $n$ singletons we study the collision spectrum $(X_{n,k}:2\le k\le n)$, where $X_{n,k}$ counts, throughout the history of the process, the number of collisions involving exactly $k$ blocks. Our focus is on the large $n$ asymptotics of the joint distribution of the $X_{n,k}$'s, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similarly to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure $Λ$ in the vicinity of $0$. In particular, for beta$(a,b)$-coalescents different types of limit distributions occur depending on whether $0 2$.

math.PR↗
arXiv2016

Spherically Symmetric Random Permutations

We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group $S_n$ and are consistent as $n$ varies. The extreme infinitely spherically symmetric permutation-valued processes are identified for the Hamming, Kendall-tau and Caley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity.

math.PR↗
arXiv2016

Guess the Larger Number

We discuss variations of the zero-sum game where Bob selects two distinct numbers, and Alice learns one of them to make a guess which of the numbers is the larger.

math.HO↗
arXiv2014

Record-dependent measures on the symmetric groups

A probability measure $P_n$ on the symmetric group ${\mathfrak S}_n$ is said to be record-dependent if $P_n(σ)$ depends only on the set of records of a permutation $σ\in{\mathfrak S}_n$. A sequence $P=(P_n)_{n\in{\mathbb N}}$ of consistent record-dependent measures determines a random order on $\mathbb N$. In this paper we describe the extreme elements of the convex set of such $P$. This problem turns out to be related to the study of asymptotic behavior of permutation-valued growth processes, to random extensions of partial orders, and to the measures on the Young-Fibonacci lattice.

math.PR↗
arXiv2012

On asymptotics of the beta-coalescents

We show that the total number of collisions in the exchangeable coalescent process driven by the beta $(1,b)$ measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance $b=1$, which corresponds to the Bolthausen--Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta $(a,b)$-coalescents with $0<a<1$ leads to a simplified derivation of the known $(2-a)$-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta $(1,b)$-coalescent by exploiting the method of sequential approximations.

math.PR↗
arXiv2012

Regenerative compositions in the case of slow variation: A renewal theory approach

A regenerative composition structure is a sequence of ordered partitions derived from the range of a subordinator by a natural sampling procedure. In this paper, we extend previous studies Barbour and Gnedin (2006), Gnedin, Iksanov and Marynych (2010) and Gnedin, Pitman and Yor (2006) on the asymptotics of the number of blocks $K_n$ in the composition of integer $n$, in the case when the L{é}vy measure of the subordinator has a property of slow variation at 0. Using tools from the renewal theory the limit laws for $K_n$ are obtained in terms of integrals involving the Brownian motion or stable processes. In other words, the limit laws are either normal or other stable distributions, depending on the behavior of the tail of L{é}vy measure at $\infty$. Similar results are also derived for the number of singleton blocks.

math.PR↗
arXiv2012

Block characters of the symmetric groups

Block character of a finite symmetric group S(n) is a positive definite function which depends only on the number of cycles in permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the characters to descent representations and the coinvariant algebra of S(n). The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group S(\infty) along with their connection to the Thoma characters of the infinite linear group GL(\infty,q) over a Galois field.

math.RT↗
arXiv2012

Exponential-Uniform Identities Related to Records

We consider a rectangular grid induced by the south-west records from the planar Poisson point process in $R^2_+$. A random symmetry property of the matrix whose entries are the areas of tiles of the grid implies cute multivariate distributional identities for certain rational functions of independent exponential and uniform random variables.

math.PR↗
arXiv2012

A generalization of the Erdős-Turán law for the order of random permutation

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős-Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM$(θ)$ distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

math.PR↗
arXiv2011

Moments of random sums and Robbins' problem of optimal stopping

Robbins' problem of optimal stopping asks one to minimise the expected {\it rank} of observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the {\it value} of the stopped variable under the rule optimal in the sense of the rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.

math.PR↗
arXiv2011

The Unlucky Door

In the famous Three-Door-Game Monte cannot help to win all the time by signaling location of the prize, using only the freedom he allowed to use. No matter which strategies played, there is always at least one door where the prize will not be found. However, already in the game with four doors cooperative Monte can reveal two useless doors in sequence (leaving two doors unrevealed), to inform Conie about location of the prize, so enabling her to beat the only-switching strategies and win all the time.

math.HO↗
arXiv2011

The Monty Hall Problem in the Game Theory Class

The basic Monty Hall problem is explored to introduce into the fundamental concepts of the game theory and to give a complete Bayesian and a (noncooperative) game-theoretic analysis of the situation. Simple combinatorial arguments are used to exclude the holding action and to find minimax solutions.

math.HO↗
arXiv2011

The Doors

We emphasize the dominance in the Monty Hall problem, both in the classical scenario and its multi-door generalization. This is used to show optimality of the class of always-switching strategies for nonuniform allocation of the prize and arbitrary door-revealing mechanism in the event of match.

math.HO↗
arXiv2011

Dominance in the Monty Hall Problem

Elementary decision-theoretic analysis of the Monty Hall dilemma shows that the problem has dominance. This makes possible to discard nonswitching strategies, without making any assumptions on the prior distribution of factors out of control of the decision maker. A path to the Bayesian and the minimax decision-making environments is then straightforward.

math.HO↗
arXiv2011

The two-sided infinite extension of the Mallows model for random permutations

We introduce a probability distribution Q on the group of permutations of the set Z of integers. Distribution Q is a natural extension of the Mallows distribution on the finite symmetric group. A one-sided infinite counterpart of Q, supported by the group of permutations of the set N of natural numbers, was studied previously in our paper [Gnedin and Olshanski, Ann. Prob. 38 (2010), 2103-2135; arXiv:0907.3275]. We analyze various features of Q such as its symmetries, the support, and the marginal distributions.

math.PR↗