Showing papers on "Central limit theorem published in 1998"

Decoupling: From Dependence to Independence

Abstract: 1 Sums of Independent Random Variables.- 1.1 Levy-Type Maximal Inequalities.- 1.2 Hoffmann-J?rgensen Type Inequalities.- 1.3 The Khinchin-Kahane Inequalities.- 1.4 Moment Bounds.- 1.4.1 Maxima.- 1.4.2 Estimating La-Norms in Hilbert Space.- 1.4.3 K-Function Bounds.- 1.4.4 A General Decoupling for Sums of Arbitrary Positive Random Variables and Martingales.- 1.5 Estimates with Sharp Constants for the La-Norms of Sums of Independent Random Variables: The L-Function.- 1.6 References for Chapter 1.- 2 Randomly Stopped Processes With Independent Increments.- 2.1 Wald's Equations.- 2.2 Good-Lambda Inequalities.- 2.3 Randomly Stopped Sums of Independent Banach-Valued Variables.- 2.4 Proof of the Lower Bound of Theorem 2.3.1.- 2.5 Continuous Time Processes.- 2.6 Burkholder-Gundy Type Inequalities in Banach Spaces.- 2.7 From Boundary Crossing of Nonrandom Functions to First Passage Times of Processes with Independent Increments.- 2.8 References for Chapter 2.- 3 Decoupling of U-Statistics and U-Processes.- 3.1 Decoupling of U-Processes: Convex Functions.- 3.2 Hypercontractivity of Rademacher Chaos Variables.- 3.3 Minorization of Tail Probabilities: The Paley-Zygmund Argument and a Conditional Jensen's Inequality.- 3.4 Decoupling of U-processes: Tail Probabilities.- 3.5 Randomization136.- 3.5.1 Moment Inequalities for Randomized U-Statistics139.- 3.5.2 Randomization of Tail Probabilities for U-Statistics and Processes.- 3.6 References for Chapter 3.- 4 Limit Theorems for U-Statistics.- 4.1 Some Inequalities the Law of Large Numbers.- 4.1.1 Hoffmann-J?rgensen's Inequality for U-Processes.- 4.1.2 An Application to the Law of Large Numbers.- 4.1.3 Exponential Inequalities for Canonical U-Statistics.- 4.2 Gaussian Chaos and the Central Limit Theorem for Canonical U-Statistics.- 4.3 The Law of the Iterated Logarithm for Canonical U-Statistics.- 4.4 References for Chapter 4.- 5 Limit Theorems for U-Processes.- 5.1 Some Background on Asymptotics of Processes, Metric Entropy, and Vapnik-?ervonenkis Classes of Functions: Maximal Inequalities.- 5.1.1 Convergence in Law of Sample Bounded Processes.- 5.1.2 Maximal Inequalities Based on Metric Entropy.- 5.1.3 Vapnik-?ervonenkis Classes of Functions.- 5.2 The Law of Large Numbers for U-Processes.- 5.3 The Central Limit Theorem for U-Processes.- 5.4 The Law of the Iterated Logarithm for Canonical U-Processes.- 5.4.1 The Bounded LIL.- 5.4.2 The Compact LIL.- 5.5 Statistical Applications.- 5.5.1 The Law of Large Numbers for the Simplicial Median.- 5.5.2 The Central Limit Theorem for the Simplicial Median.- 5.5.3 Truncated Data.- 5.6 References for Chapter 5.- 6 General Decoupling Inequalities for Tangent Sequences.- 6.1 Some Definitions and Examples.- 6.2 Exponential Decoupling Inequalities for Sums.- 6.3 Tail Probability andLpInequalities for Tangent Sequences I.- 6.4 Tail Probability and Moment Inequalities for Tangent Sequences II: Good-Lambda Inequalities.- 6.5 Differential Subordination and Applications.- 6.6 Decoupling Inequalities Compared to Martingale Inequalities.- 6.7 References for Chapter 6323.- 7 Conditionally Independent Sequences.- 7.1 The Principle of Conditioning and Related Results.- 7.2 Analysis of a Sequence of Two-by-Two Tables.- 7.3 SharpLpComparison of Sums of Arbitrarily Dependent Variables to Sums of CI Variables.- 7.4 References for Chapter 7.- 8 Further Applications of Decoupling.- 8.1 Randomly Stopped Canonical U-Statistics.- 8.1.1 Wald's Equation for Canonical U-Statistics.- 8.1.2 Moment Bounds for Regular and Randomly StoppedU-Statistics.- 8.1.3 Moment Convergence in Anscombe's Theorem forU-Statistics.- 8.2 A General Class of Exponential Inequalities for Martingales and Ratios.- 8.3 References for Chapter 8.- References.

Balls and bins: a study in negative dependence

Abstract: This paper investigates the notion of negative dependence amongst random variables and attempts to advocate its use as a simple and unifying paradigm for the analysis of random structures and algorithms. The assumption of independence between random variables is often very convenient for the several reasons. Firstly, it makes analyses and calculations much simpler. Secondly, one has at hand a whole array of powerful mathematical concepts and tools from classical probability theory for the analysis, such as laws of large numbers, central limit theorems and large deviation bounds which are usually derived under the assumption of independence. Unfortunately, the analysis of most randomized algorithms involves random variables that are not independent. In this case, classical tools from standard probability theory like large deviation theorems, that are valid under the assumption of independence between the random variables involved, cannot be used as such. It is then necessary to determine under what conditions of dependence one can still use the classical tools. It has been observed before [32, 33, 38, 8], that in some situations, even though the variables involved are not independent, one can still apply some of the standard tools that are valid for independent variables (directly or in suitably modified form), provided that the variables are dependent in specific ways. Unfortunately, it appears that in most cases somewhat ad hoc strategems have been devised, tailored to the specific situation at hand, and that a unifying underlying theory that delves deeper into the nature of dependence amongst the variables involved is lacking. A frequently occurring scenario underlying the analysis of many randomised algorithms and processes involves random variables that are, intuitively, dependent in the following negative way: if one subset of the variables is "high" then a disjoint subset of the variables is "low". In this paper, we bring to the forefront and systematize some precise notions of negative dependence in the literature, analyse their properties, compare them relative to each other, and illustrate them with several applications. One specific paradigm involving negative dependence is the classical "balls and bins" experiment. Suppose we throw m balls into n bins independently at random. For i in [n], let Bi be the random variable denoting the number of balls in the ith bin. We will often refer to these variables as occupancy numbers. This is a classical probabilistic paradigm [16, 22, 26] (see also [31, sec. 3.1]) that underlies the analysis of many probabilistic algorithms and processes. In the case when the balls are identical, this gives rise to the well-known multinomial distribution [16, sec VI.9]: there are m repeated independent trials (balls) where each trial (ball) can result in one of the outcomes E1, ..., En (bins). The probability of the realisation of event Ei is pi for i in [n] for each trial. (Of course the probabilities are subject to the condition Sum_i pi = 1.) Under the multinomial distribution, for any integers m1, ..., mn such that Sum_i mi = m the probability that for each i in [n], event Ei occurs mi times is m! m1! : : :mn!pm1 1 : : :pmn n : The balls and bins experiment is a generalisation of the multinomial distribution: in the general case, one can have an arbitrary set of probabilities for each ball: the probability that ball k goes into bin i is pi;k, subject only to the natural restriction that for each ball k, P i pi;k = 1. The joint distribution function correspondingly has a more complicated form. A fundamental natural question of interest is: how are these Bi related? Note that even though the balls are thrown independently of each other, the Bi variables are not independent; in particular, their sum is fixed to m. Intuitively, the Bi's are negatively dependent on each other in the manner described above: if one set of variables is "high", a disjoint set is "low". However, establishing such assertions precisely by a direct calculation from the joint distribution function, though possible in principle, appears to be quite a formidable task, even in the case where the balls are assumed to be identical. One of the major contributions of this paper is establishing that the the Bi are negatively dependent in a very strong sense. In particular, we show that the Bi variables satisfy negative association and negative regression, two strong notions of negative dependence that we define precisely below. All the intuitively obvious assertions of negative dependence in the balls and bins experiment follow as easy corollaries. We illustrate the usefulness of these results by showing how to streamline and simplify many existing probabilistic analyses in literature.

On Convergence Rates in the Central Limit Theorems for Combinatorial Structures

Abstract: Flajolet and Soria established several central limit theorems for the parameter `number of components? in a wide class of combinatorial structures. In this paper, we shall prove a simple theorem which applies to characterize the convergence rates in their central limit theorems. This theorem is also applicable to arithmetical functions. Moreover, asymptotic expressions are derived for moments of integral order. Many examples from different applications are discussed.

A statistical basis for lognormal shadowing effects in multipath fading channels

Abstract: Empirical justifications for the lognormal, Rayleigh and Suzuki (1977) probability density functions in multipath fading channels are examined by quantifying the rates of convergence of the central limit theorem (CLT) for the addition and multiplication of random variables. The accuracy of modeling the distribution of rays which experience multiple reflections/diffractions between transmitter and receiver as lognormal is quantified. In addition, it is shown that the vector sum of lognormal rays, such as in a narrow-band signal envelope, may best be approximated as being either Rayleigh, lognormal or Suzuki distributed depending on the fading channel conditions. These conditions are defined statistically.

A central limit theorem for stationary random fields

Abstract: We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields.

A central-limit-theorem-based approach for analyzing queue behavior in high-speed networks

Abstract: In this paper, we study P(/spl Qscr/>x), the tail of the steady-state queue length distribution at a high-speed multiplexer. In particular, we focus on the case when the aggregate traffic to the multiplexer can be characterized by a stationary Gaussian process. We provide two asymptotic upper bounds for the tail probability and an asymptotic result that emphasizes the importance of the dominant time scale and the maximum variance. One of our bounds is in a single-exponential form and can be used to calculate an upper bound to the asymptotic constant. However, we show that this bound, being of a single-exponential form, may not accurately capture the tail probability. Our asymptotic result on the importance of the maximum variance and our extensive numerical study on a known lower bound motivate the development of our second asymptotic upper bound. This bound is expressed in terms of the maximum variance of a Gaussian process, and enables the accurate estimation of the tail probability over a wide range of queue lengths. We apply our results to Gaussian as well as multiplexed non-Gaussian input sources, and validate their performance via simulations. Wherever possible, we have conducted our simulation study using importance sampling in order to improve its reliability and to effectively capture rare events. Our analytical study is based on extreme value theory, and therefore different from the approaches using traditional Markovian and large deviations techniques.

State-dependent stochastic networks. Part I. Approximations and applications with continuous diffusion limits

Abstract: In a state-dependent queueing network, arrival and service rates, as well as routing probabilities, depend on the vector of queue lengths. For properly normalized such networks, we derive functional laws of large numbers (FLLNs) and functional central limit theorems (FCLTs). The former support fluid approximations and the latter support diffusion refinements. The fluid limit in FLLN is the unique solution to a multidimensional autonomous ordinary differential equation with state-dependent reflection. The diffusion limit in FCLT is the unique strong solution to a stochastic differential equation with time-dependent reflection. Examples are provided that demonstrate how such approximations facilitate the design, analysis and optimization of various manufacturing, service, communication and other systems.

On Pattern Frequency Occurrences in a Markovian Sequence

Abstract: Consider a given pattern H and a random text T generated by a Markovian source. We study the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately. We present exact and asymptotic formulae for moments (including the variance), and probability of r pattern occurrences for three different regions of r , namely: (i) r=O(1) , (ii) central limit regime, and (iii) large deviations regime. In order to derive these results, we first construct certain language expressions that characterize pattern occurrences which are later translated into generating functions. We then use analytical methods to extract asymptotic behaviors of the pattern frequency from the generating functions. These findings are of particular interest to molecular biology problems (e.g., finding patterns with unexpectedly high or low frequencies, and gene recognition), information theory (e.g., second-order properties of the relative frequency), and pattern matching algorithms (e.g., q -gram algorithms).

Reconstructing Nonlinear Stochastic Bias from Velocity Space Distortions

Abstract: We propose a strategy to measure the dark matter power spectrum using minimal assumptions about the galaxy distribution and the galaxy-dark matter cross-correlations. We argue that on large scales the central limit theorem generically assures Gaussianity of each smoothed density field, but not coherence. Asymptotically, the only surviving parameters on a given scale are galaxy variance σ, bias b = Ω0.6/β, and the galaxy-dark matter cross-correlation coefficient r. These can all be determined by measuring the quadrupole and hexadecapole velocity distortions in the power spectrum. Measuring them simultaneously may restore consistency between all β-determinations independent of galaxy type. The leading deviations from Gaussianity are conveniently parameterized by an Edgeworth expansion. In the mildly nonlinear regime, two additional parameters describe the full picture: the skewness parameter s and nonlinear bias b2. They can both be determined from the measured skewness combined with second-order perturbation theory or from an N-body simulation. By measuring the redshift distortion of the skewness, one can measure the density parameter Ω with minimal assumptions about the galaxy formation process. This formalism also provides a convenient parameterization to quantify statistical galaxy formation properties.

On the blockwise bootstrap for empirical processes for stationary sequences

Abstract: In this paper, we study the weak convergence to an appropriate Gaussian process of the empirical process of the block-based bootstrap estimator proposed by Kunsch for stationary sequences. The classes of processes investigated are weak dependent and associated sequences. We also prove that, differently from the independent situation, the bootstrapped estimator of the mean of certain dependent sequences satisfies the central limit theorem while the mean of the original sequence does not.

Results and problems related to the pointwise central limit theorem

Abstract: Publisher Summary The discovery of the pointwise central limit theorem created a considerable interest in logarithmic limit theorems and in the past decade a large number of interesting results of this type are proved The purpose of this chapter is to survey some basic theorems in the field and to discuss open problems

Central limit and functional central limit theorems for hilbert-valued dependent heterogeneous arrays with applications

Abstract: We obtain new central limit theorems (CLT's) and functional central limit theorems (FCLT's) for Hilbert-valued arrays near epoch dependent on mixing processes, and also new FCLT's for general Hilbert-valued adapted dependent heterogeneous arrays. These theorems are useful in delivering asymptotic distributions for parametric and nonparametric estimators and their functionals in time series econometrics. We give three significant applications for near epoch dependent observations: (1) A new CLT for any plug-in estimator of a cumulative distribution function (c.d.f.) (e.g., an empirical c.d.f., or a c.d.f. estimator based on a kernel density estimator), which can in turn deliver distribution results for many Von Mises functionals; (2) a new limiting distribution result for degenerate U -statistics, which delivers distribution results for Bierens's integrated conditional moment tests; (3) a new functional central limit result for Hilbert-valued stochastic approximation procedures, which delivers distribution results for nonparametric recursive generalized method of moment estimators, including nonparametric adaptive learning models.

Asymptotics of Spectral Projections of Some Random Matrices Approximating Integral Operators

Abstract: Given a probability space (S, S, P) and a symmetric measurable real valued kernel h on S × S, which defines a compact integral operator H from L 2(P) into L 2(P), we study the asymptotic behavior of spectral projections of the random n × n matrices H n with entries n −1h(X i,X j), 1 ≤ i,j ≤ n, and H n, obtained from H n by deleting of its diagonal. Here (X 1,… , X n) is a sample of independent random variables in (S, S) with common distribution P. We show that if H is a Hilbert-Schmidt operator, then the spectral projections of H n converge a.s. to the spectral projections of H (the convergence is understood in the sense of quadratic forms). Under slightly stronger assumptions, the convergence also holds for the spectral projections of H n. Moreover, under much stronger assumptions on the kernel h, we show that the fluctuations of the spectral projections of H n or H n are asymptotically Gaussian.

Maximum of partial sums and an invariance principle for a class of weak dependent random variables

Abstract: The aim of this paper is to investigate the properties of the maximum of partial sums for a class of weakly dependent random variables which includes the instantaneous filters of a Gaussian sequence having a positive continuous spectral density. The results are used to obtain an invariance principle and the convergence of the moments in the central limit theorem.

Zipf's law is not a consequence of the central limit theorem

Abstract: It has been observed that the rank statistics of string frequencies of many symbolic systems (e.g., word frequencies of natural languages) follows Zipf's law in good approximation. We show that, contrary to claims in the literature, Zipf's law cannot be realized by the central limit theorem(s). The observation that a log-normal distribution of string frequencies yields an approximately Zipf-like rank statistics is actually misleading. Indeed, Zipf's law for the rank statistics is strictly equivalent to a power law distribution of frequencies. There are two natural ways to perform the infinite size limit for the vocabulary. The first one is the method of choice in the literature; it makes the upper word length bound tend to infinity and leads in the case of a multistate Bernoulli process via a central limit theorem to a log-normal frequency distribution. An alternative and for text samples actually better realizable way is to make the lower frequency bound tend to zero. This limit procedure leads to a power law distribution and hence to Zipf's law---at least for Bernoulli processes and to a very good approximation for natural languages where it passes the ${\ensuremath{\chi}}^{2}$ test. For the Bernoulli case we will give a heuristic proof.

Notions of Independence Related to the Free Group

Abstract: The central limit problem for algebraic probability spaces associated with the Haagerup states on the free group with countably many generators leads to a new form of statistical independence in which the singleton condition is not satisfied. This circumstance allows us to obtain nonsymmetric distributions from the central limit theorems deduced from this notion of independence. In the particular case of the Haagerup states, the role of the Gaussian law is played by the Ullman distribution. The limit process is explicitly realized on the finite temperature Boltzmannian Fock space. The role of entangled ergodic theorems in the proof of the central limit theorems is discussed.

Central Limit Theorem for the Adjacency Operators on the Infinite Symmetric Group

Abstract: An adjacency operator on a group is a formal sum of (left) regular representations over a conjugacy class. For such adjacency operators on the infinite symmetric group which are parametrized by the Young diagrams, we discuss the correlation of their powers with respect to the vacuum vector state. We compute exactly the correlation function under suitable normalization and through the infinite volume limit. This approach is viewed as a central limit theorem in quantum probability, where the operators are interpreted as random variables via spectral decomposition. In [K], Kerov showed the corresponding result for one-row Young diagrams. Our formula provides an extension of Kerov's theorem to the case of arbitrary Young diagrams.

Fluctuations of a Surface Submitted to a Random Average Process

Abstract: We consider a hypersurface of dimension $d$ imbedded in a $d+1$ dimensional space. For each $x\in Z^d$, let $\eta_t(x)\in R$ be the height of the surface at site $x$ at time $t$. At rate $1$ the $x$-th height is updated to a random convex combination of the heights of the `neighbors' of $x$. The distribution of the convex combination is translation invariant and does not depend on the heights. This motion, named the random average process (RAP), is one of the linear processes introduced by Liggett (1985). Special cases of RAP are a type of smoothing process (when the convex combination is deterministic) and the voter model (when the convex combination concentrates on one site chosen at random). We start the heights located on a hyperplane passing through the origin but different from the trivial one $\eta(x)\equiv 0$. We show that, when the convex combination is neither deterministic nor concentrating on one site, the variance of the height at the origin at time $t$ is proportional to the number of returns to the origin of a symmetric random walk of dimension $d$. Under mild conditions on the distribution of the random convex combination, this gives variance of the order of $t^{1/2}$ in dimension $d=1$, $\log t$ in dimension $d=2$ and bounded in $t$ in dimensions $d\ge 3$. We also show that for each initial hyperplane the process as seen from the height at the origin converges to an invariant measure on the hyper surfaces conserving the initial asymptotic slope. The height at the origin satisfies a central limit theorem. To obtain the results we use a corresponding probabilistic cellular automaton for which similar results are derived. This automaton corresponds to the product of (infinitely dimensional) independent random matrices whose rows are independent.

A new inductive approach to the lace expansion for self-avoiding walks

Abstract: We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤ d where loops of length m are penalised by a factor e −β/m p (0 4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0.

The lognormal distribution of software failure rates: origin and evidence

Abstract: An understanding of the distribution of software failure rates and its origin will strengthen the relation of software reliability engineering both to other aspects of software engineering and to the wider field of reliability engineering. The paper proposes that the distribution of failure rates for faults in software systems tends to be lognormal. Many successful analytical models of software behavior share assumptions that suggest that the distribution of software event rates will asymptotically approach lognormal. The lognormal distribution has its origin in the complexity, that is the depth of conditionals, of software systems and the fact that event rates are determined by an essentially multiplicative process. The central limit theorem links these properties to the lognormal: just as the normal distribution arises when summing many random terms, the lognormal distribution arises when the value of a variable is determined by the multiplication of many random factors. Because the distribution of event rates tends to be lognormal and faults are just a random subset or sample of the events, the distribution of the failure rates of the faults also tends to be lognormal. Failure rate distributions observed by other researchers in twelve repetitive-run experiments and nine sets of field failure data are analyzed and demonstrated to support the lognormal hypothesis. The ability of the lognormal to fit these empirical failure rate distributions is superior to that of the gamma distribution (the basis of the Gamma/EOS family of reliability growth models) or a Power-law model.

Limit theorems for discretely observed stochastic volatility models

Abstract: A general set-up is proposed to study stochastic volatility models. We consider here a two-dimensional diffusion process ( Y t,V t) and assume that only ( Y t) is observed at n discrete times with regular sampling interval Δ . The unobserved coordinate ( V t) is an ergodic diffusion which rules the diffusion coefficient (or volatility) of ( Y t) . The following asymptotic framework is used: the sampling interval tends to 0 , while the number of observations and the length of the observation time tend to infinity. We study the empirical distribution associated with the observed increments of ( Y t) . We prove that it converges in probability to a variance mixture of Gaussian laws and obtain a central limit theorem. Examples of models widely used in finance, and included in this framework, are given.

A multivariate central limit theorem for continuous local martingales

Abstract: A theorem on the weak convergence of a properly normalized multivariate continuous local martingale is proved. The time-change theorem used for this purpose allows for short and transparent arguments.

New Examples of Convolutions and Non-Commutative Central Limit Theorems

Abstract: A family of transformations on the set of all probability measures on the real line is introduced, which makes it possible to define new examples of convolutions. The associated central limit theorems are studied, and examples of the limit measures, related to the classical, free and boolean convolutions, are shown.

A central limit theorem for contractive stochastic dynamical systems

Abstract: If ( F n ) n ∈ℕ is a sequence of independent and identically distributed random mappings from a second countable locally compact state space 𝕏 to 𝕏 which itself is independent of the 𝕏-valued initial variable X 0 , the discrete-time stochastic process ( X n ) n ≥0 , defined by the recursion equation X n = F n ( X n −1 ) for n ∈ℕ, has the Markov property. Since 𝕏 is Polish in particular, a complete metric d exists. The random mappings ( F n ) n ∈ℕ are assumed to satisfy ℙ-a.s. Conditions on the distribution of l ( F n ) are given for the existence of an invariant distribution of X 0 making the process ( X n ) n ≥0 stationary and ergodic. Our main result corrects a central limit theorem by Łoskot and Rudnicki (1995) and removes an error in its proof. Instead of trying to compare the sequence φ ( X n ) n ≥0 for some φ : 𝕏 → ℝ with a triangular scheme of independent random variables our proof is based on an approximation by a martingale difference scheme.

Anomalous transport in a model of hamiltonian round-off

Abstract: We study the propagation of round-off error near the periodic orbits of a linear area-preserving map - a planar rotation by a rational angle - which is discretized on a lattice in such a way as to retain invertibility. We consider the round-off error probability distribution as a function of time t, and we show that for each t this is an algebraic number, which can be calculated exactly. We prove that its kth moment increases asymptotically as , where is the fractional dimension of a self-similar set related to periodic orbits of long-period, while G is a bounded function, periodic in the logarithm of t. This implies the diffusion coefficient displays bounded variations, while all higher order transport coefficients diverge, resulting in anomalous transport. This result contrasts with the case of irrational rotations, where the existence of a central limit theorem has been recently established (Vladimirov I 1996 Preprint Deakin University).

Functional central limit theorem for the empirical process of short memory linear processes

Abstract: We prove the functional central limit theorem for the empirical distribution function of a stationary causal moving average sequence with absolutely summable weights.