Triangular Arrays Of Random Variables Research Articles

The Convergence Rate of Option Prices in Trinomial Trees

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1/n and 1/n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations.

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

Exponential tail estimates in the law of ordinary logarithm (LOL) for triangular arrays of random variables

On the converse law of large numbers

Given a triangular array with mn random variables in the nth row and a growth rate {kn}n=1∞ with lim sup n→∞(kn/mn)<1, if the empirical distributions converge for any subarrays with the same growth rate, then the triangular array is asymptotically independent. In other words, if the empirical distribution of any kn random variables in the nth row of the triangular array is asymptotically close in probability to the law of a randomly selected random variable among these kn random variables, then two randomly selected random variables from the nth row of the triangular array are asymptotically close to being independent. This provides a converse law of large numbers by deriving asymptotic independence from a sample stability condition. It follows that a triangular array of random variables is asymptotically independent if and only if the empirical distributions converge for any subarrays with a given asymptotic density in (0,1). Our proof is based on nonstandard analysis, a general method arisen from mathematical logic, and Loeb measure spaces in particular.

Stochastic Bernstein polynomials: uniform convergence in probability with rates

We introduce stochastic variants of the classical Bernstein polynomials associated with a continuous function f, built up from a general triangular array of random variables. We discuss the uniform convergence in probability of the approximation process that they represent, providing at the same time rates of convergence. In the particular case in which the triangular array of random variables consists of the uniform order statistics, we give a positive answer to a conjectured raised in Wu and Zhou (Adv. Comput. Math. 46, 8, 2020) about an exponential rate of convergence in probability.

A weak convergence result for sequential empirical processes under weak dependence

ABSTRACTThe purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying α-mixing triangular array of random variables. In particular, the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore, under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as possible applications.

A consistent bootstrap procedure for the maximum score estimator

In this paper we propose a new model-based smoothed bootstrap procedure for making inference on the maximum score estimator of Manski (1975, 1985) and prove its consistency. We provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem. We compare the finite sample performance of different bootstrap procedures through simulation studies. The results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes, including the m-out-of-n bootstrap. Additionally, we prove a convergence theorem for triangular arrays of random variables arising from binary choice models, which may be of independent interest.

Should annual maximum temperatures follow a generalized extreme value distribution?

Motivated by the study of annual temperature extremes, two new results on the limiting distribution of block maxima of random variables with varying upper bounds are obtained. One gives a generalized extreme value distribution as the limit, but with a different shape parameter from that obtained when the bound on the random variables does not vary. The other gives a limiting distribution that is only a generalized extreme value in certain cases. Both results consider triangular arrays of random variables in order to mimic the property of an upper bound that changes slowly with the day of the year, as seems to occur for temperature data at many locations. An analysis of 140 years of daily temperatures in New York City shows mixed results in terms of the ability of the theory presented here to provide new insights into the behaviour of extreme temperatures at this location.

Limiting behavior for arrays of row-wise upper extended negatively dependent random variables

Our aim is to obtain deterministic bounds for the row sum elements of a random triangular array introducing, thereunto, a dependence structure for triangular arrays of random variables which extend the concepts of upper and lower extended negatively dependence already known for random variables. Concretely, for triangular arrays \({\{X_{n,k}, 1 \leqq k \leqq n, n \geqq 1\}}\) of row-wise upper extended negatively dependent random variables with dominating sequence \({\{M_{n}, n \geqq 1\}}\) weakly mean dominated by a random variable X and sequences \({\{b_{n}\}}\) of positive constants, conditions are stated to ensure the deterministic boundedness of \({\Sigma^{n}_{k=1}(X_{n,k} - {\mathbb E}X_{n,k})/\sqrt{b_{n} {\rm Log} n}}\), where \({{\rm Log} n := {\rm log} {\rm max}\{n, e\}}\). In particular, a sufficient moment condition is given permitting us to achieve our goal under the rate of the so called “Law of the Logarithm”.

An isometric study of the Lindeberg–Feller central limit theorem via Stein’s method

We use Stein’s method to prove a generalization of the Lindeberg–Feller CLT providing an upper and a lower bound for the superior limit of the Kolmogorov distance between a normally distributed random variable and the rowwise sums of a rowwise independent triangular array of random variables which is asymptotically negligible in the sense of Feller. A natural example shows that the upper bound is of optimal order. The lower bound improves a result by Andrew Barbour and Peter Hall.

A note on the ASCLT for triangular arrays of random variables with an extension to U-statistics

In this paper, we obtain an almost sure central limit theorem for products of independent sums of positive random variables. An extension of the result gives an ASCLT for the U-statistics.

A law of the single logarithm for weighted sums of arrays applied to bootstrap model selection in regression

We generalize a law of the single logarithm obtained by Qi (1994) and Li et al. (1995) to the case of weighted sums of triangular arrays of random variables. We apply this result to bootstrapping the all-subsets model selection problem in regression, where we show that the popular Bayesian Information Criterion of Schwarz (1978) is no longer asymptotically consistent.

Large deviation principles for sequences of logarithmically weighted means

In this paper we consider several examples of sequences of partial sums of triangular arrays of random variables { X n : n ⩾ 1 } ; in each case X n converges weakly to an infinitely divisible distribution (a Poisson distribution or a centered Normal distribution). For each sequence we prove large deviation results for the logarithmically weighted means { 1 log n ∑ k = 1 n 1 k X k : n ⩾ 1 } with speed function v n = log n . We also prove a sample path large deviation principle for { X n : n ⩾ 1 } defined by X n ( ⋅ ) = ∑ i = 1 n U i ( σ 2 ⋅ ) n , where σ 2 ∈ ( 0 , ∞ ) and { U n : n ⩾ 1 } is a sequence of independent standard Brownian motions.

Large Deviations for Symmetrised Empirical Measures

In this paper we prove a Large Deviation Principle for the sequence of symmetrised empirical measures \(\frac{1}{n}\sum_{i=1}^{n}\delta_{(X^{n}_{i},X^{n}_{\sigma_{n}(i)})}\) where σn is a random permutation and ((Xin)1≤i≤n)n≥1 is a triangular array of random variables with suitable properties. As an application we show how this result allows to improve the Large Deviation Principles for symmetrised initial-terminal conditions bridge processes recently established by Adams, Dorlas and Konig.

Condition for the Convergence of Maxima of Random Triangular Arrays

We consider the convergence of the maxima of a triangular array of random variables. Sufficient and necessary conditions are discussed, assuming that the underlying distributions are twice differentiable. Our results extend those known for the iid. case.

A central limit theorem for exchangeable random variables on a locally compact abelian group

A central limit theorem is given for uniformly infinitesimal triangular arrays of random variables in which the random variables in each row are exchangeable and take values in a locally compact second countable abelian group. The limiting distribution in the theorem is Gaussian.

On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form ∑ j=1 n W njX nj where { W nj , 1⩽ j⩽ n, n⩽1} and { X nj , 1⩽ j⩽ n, n⩾1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.

A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN

The aim of this paper is to provide a viable measure–theoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus constitute the `right' class for general stochastic modeling. The primary concern of the paper is to characterize those hyperfinite processes satisfying the exact law of large numbers by using the basic notions of conditional expectation, orthogonality, uncorrelatedness and independence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed via a biorthogonal expansion of the Karhunen–Loéve type and via the representation in terms of the simpler hyperfinite Loeb counting spaces. A universality property for atomless Loeb product spaces is formulated to show the abundance of processes satisfying the law. Generalizations to a hyperfinite number of continuous (or discrete) parameter stochastic processes are considered. The various necessary and sufficient conditions for the validity of the law provide a rather complete understanding about the cancelation of individual risks or uncertainty in general settings. Some explicit asymptotic interpretations are also given.

Limit theorems for random symmetric functions

In this paper, based on theorems for limit distributions of empirical power processes for the i.i.d. case and for the case with independent triangular arrays of random variables, we prove limit theorems for U- and V-statistics determined by generalized polynomial kernel functions. We also show that under some natural conditions the limit distributions can be represented as functionals on the limit process of the normed empirical power process. We consider the one-sample case, as well as multi-sample cases.

Sudakov's typical marginals, random linear functionals and a conditional central limit theorem

V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure $P$ and a random (a.s.) linear functional $F$ on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of $P$ under $F$ are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.

On convergence to the poisson law for distributions of the sum of random variables connected into a chain

This paper deals with the triangular array of random variables where every sequence is connected into a stationary Markov chain. For the uniformly strongly mixing chain we find conditions under which the distributions of the sum converge to the Poisson law. A more general case of theL p-regular Markov chain is also considered.