Let $xi_1,xi_2,ldots$ be a sequence of independent copies of a random vector in $mathbb R^d$ having an absolutely continuous distribution. Consider a random walk $S_i:=xi_1+cdots+xi_i$, and let $C_{n,d}:=text{conv}(0,S_1,S_2,ldots,S_n)$ be the convex
hull of the first $n+1$ points it has visited. The polytope $C_{n,d}$ is called $k$-neighborly if for every indices $0leq i_0 <cdots < i_kleq n$ the convex hull of the $k+1$ points $S_{i_0},ldots, S_{i_k}$ is a $k$-dimensional face of $C_{n,d}$. We study the probability that $C_{n,d}$ is $k$-neighborly in various high-dimensional asymptotic regimes, i.e. when $n$, $d$, and possibly also $k$ diverge to $infty$. There is an explicit formula for the expected number of $k$-dimensional faces of $C_{n,d}$ which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of $C_{n,d}$. This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.
The $K$-hull of a compact set $Asubsetmathbb{R}^d$, where $Ksubset mathbb{R}^d$ is a fixed compact convex body, is the intersection of all translates of $K$ that contain $A$. A set is called $K$-strongly convex if it coincides with its $K$-hull. We p
ropose a general approach to the analysis of facial structure of $K$-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of $k$-dimensional faces, for all $k=0,dots,d-1$. We then apply our theory in the case when $A=Xi_n$ is a sample of $n$ points picked uniformly at random from $K$. We show that in this case the set of $xinmathbb{R}^d$ such that $x+K$ contains the sample $Xi_n$, upon multiplying by $n$, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding $f$-vector of the $K$-hull of $Xi_n$ to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the $f$-vector.
Let $(xi_k,eta_k)_{kinmathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{kinmathbb{N}}$ defined by $T_k:=xi_1+cdots+xi
_{k-1}+eta_k$ for $kinmathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $jinmathbb{N}$ and $tgeq 0$, denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $leq t$. In this article we prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwells theorem and the key renewal theorem) for $N_j(t)$ under the assumption that $j=j(t)toinfty$ and $j(t)=o(t^{2/3})$ as $ttoinfty$. According to our terminology, such generations form a subset of the set of intermediate generations.
A nested occupancy scheme in random environment is a generalization of the classical Karlin infinite balls-in-boxes occupancy scheme in random environment (with random probabilities). Unlike the Karlin scheme in which the collection of boxes is uniqu
e, there is a nested hierarchy of boxes, and the hitting probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. In the present paper we assume that the random fragmentation law is given by stick-breaking in which case the infinite occupancy scheme defined by the first level boxes is known as the Bernoulli sieve. Assuming that $n$ balls have been thrown, denote by $K_n(j)$ the number of occupied boxes in the $j$th level and call the level $j$ intermediate if $j=j_ntoinfty$ and $j_n=o(log n)$ as $ntoinfty$. We prove a multidimensional central limit theorem for the vector $(K_n(lfloor j_n u_1rfloor),ldots, K_n(lfloor j_n u_ellrfloor)$, properly normalized and centered, as $ntoinfty$, where $j_ntoinfty$ and $j_n=o((log n)^{1/2})$. The present paper continues the line of investigation initiated in Buraczewski, Dovgay and Iksanov [Electron. J. Probab. 25: paper no. 123, 2020] in which the occupancy of intermediate levels $j_ntoinfty$, $j_n=o((log n)^{1/3})$ was analyzed.
A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the L{e}vy measure
of $S$ is infinite and regularly varying at zero of index $-alpha$, $alphain(0,,1)$, we find an explicit threshold $r=r(n)$, such that the number $K_{n,,r(n)}$ of blocks of size $r(n)$ converges in distribution without any normalization to a mixed Poisson distribution. The sequence $(r(n))$ turns out to be regularly varying with index $alpha/(alpha+1)$ and the mixing distribution is that of the exponential functional of $S$. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of $K_{n,,w(n)}$ in cases when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement previously known strong laws of large numbers for $K_{n,,r}$ in case of a fixed $rinmathbb{N}$. As a key tool we employ new Abelian theorems for Laplace--Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.
In this paper, we draw attention to a problem that is often overlooked or ignored by companies practicing hypothesis testing (A/B testing) in online environments. We show that conducting experiments on limited inventory that is shared between variant
s in the experiment can lead to high false positive rates since the core assumption of independence between the groups is violated. We provide a detailed analysis of the problem in a simplified setting whose parameters are informed by realistic scenarios. The setting we consider is a $2$-dimensional random walk in a semi-infinite strip. It is rich enough to take a finite inventory into account, but is at the same time simple enough to allow for a closed form of the false-positive probability. We prove that high false-positive rates can occur, and develop tools that are suitable to help design adequate tests in follow-up work. Our results also show that high false-negative rates may occur. The proofs rely on a functional limit theorem for the $2$-dimensional random walk in a semi-infinite strip.
Let $B_n(m)$ be a set picked uniformly at random among all $m$-elements subsets of ${1,2,ldots,n}$. We provide a pathwise construction of the collection $(B_n(m))_{1leq mleq n}$ and prove that the logarithm of the least common multiple of the integer
s in $(B_n(lfloor mtrfloor))_{tgeq 0}$, properly centered and normalized, converges to a Brownian motion when both $m,n$ tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of $m$ independent random variables having uniform distribution on ${1,2,ldots,n}$. Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions.
We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for t
he $limsup$ and a law of the triple logarithm for the $liminf$. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424--445]. We apply our results to several queuing systems and a birth and death process.
We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit
distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.
It is known that backward iterations of independent copies of a contractive random Lipschitz function converge almost surely under mild assumptions. By a sieving (or thinning) procedure based on adding to the functions time and space components, it i
s possible to construct a scale invariant stochastic process. We study its distribution and paths properties. In particular, we show that it is c`adl`ag and has finite total variation. We also provide examples and analyse various properties of particular sieved iterative function systems including perpetuities and infinite Bernoulli convolutions, iterations of maximum, and random continued fractions.
