Sum Theorem Research Articles

Hellinger volume and number-on-the-forehead communication complexity

Information-theoretic methods are a powerful tool in communication complexity, providing, in particular, elegant and tight lower bounds on disjointness in the number-in-the-hand (NIH) model. In this paper, we study the applicability of information theoretic methods to the multi-party number-on-the-forehead model (NOF). The lower bound on disjointness in the NIH model has two parts: a direct sum theorem and a lower bound on the one-bit AND function using a beautiful connection between Hellinger distance and protocols [4]. Inspired by this connection, we introduce the notion of Hellinger volume which lower bounds the information cost of multi-party NOF protocols. We provide tools for manipulating and proving lower bounds on Hellinger volume, and use these tools to obtain a lower bound on the information complexity of the ANDk function in the NOF setting. Finally, we discuss the difficulties of proving a direct sum theorem for information cost in the NOF model.

On Tauberian theorems for Cesàro summability of sequences of fuzzy numbers

In this paper we use the concept of the Cesaro convergence of a sequence of fuzzy numbers defined by Subrahmanyam [Cesaro summability of fuzzy real numbers, Anal, 7 (1999), 159–168] to prove some Tauberian theorems for sequences of fuzzy numbers and obtain fuzzy analogues of some classical Tauberian theorems for Cesaro summability of sequences of fuzzy numbers as a corollary.

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

In the paper, we discuss how to define long, short, and negative memories for stationary processes on ℤ and fields on ℤ d with infinite variances. We propose to distinguish the properties of dependence and memory, and to attribute memory properties not only to a stationary random process (or a field) but also to a process and the operation that we apply to this process. We deal exclusively with the summation operation, that is, we consider the limit behavior of partial sums of random processes or fields. In order to have a unified approach to processes and fields with finite and infinite variances, we propose to define memory properties via the growth of normalizing sequences in limit theorems for partial sums. Also, we propose to change a little bit the terminology: instead of terms “long and short memories,” to use positive and zero memories, respectively, leaving the term “negative memory” and introducing “strongly negative memory.” For random fields, we introduce the notions of isotropic and directional memories.

Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences

Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of , for some Laurent polynomials P(x) and Q(x) in the variable x with integer coefficients. Denoting such a sequence by , we obtain a general formula that determines the congruence class, modulo p, of the indefinite sum , for any prime p, and any positive integer r, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime p.

Weak convergence to the Student and Laplace distributions

Abstract One often observed empirical regularity is a power-law behavior of the tails of some distribution of interest. We propose a limit law for normalized random means that exhibits such heavy tails irrespective of the distribution of the underlying sampling units: the limit is a t-distribution if the random variables have finite variances. The generative scheme is then extended to encompass classic limit theorems for random sums. The resulting unifying framework has wide empirical applicability which we illustrate by considering two empirical regularities in two different fields. First, we turn to urban geography and explain why city-size growth rates are approximately t-distributed, using a model of random sector growth based on the central place theory. Second, turning to an issue in finance, we show that high-frequency stock index returns can be modeled as a generalized asymmetric Laplace process. These empirical illustrations elucidate the situations in which heavy tails can arise.

Modeling High-Frequency Non-Homogeneous Order Flows by Compound Cox Processes*

A micro-scale model is proposed for the evolution of a limit order book in modern high-frequency trading applications. Within this model, order flows are described by doubly stochastic Poisson processes (also called Cox processes) taking account of the stochastic character of the intensities of order flows. The models for the number of order imbalance (NOI) process and order flow imbalance (OFI) process are introduced as two-sided risk processes that are special compound Cox processes. These processes are sensitive indicators of the current state of the limit order book since time intervals between events in a limit order book are usually so short that price changes are relatively infrequent events. Therefore price changes provide a very coarse and limited description of market dynamics at time micro-scales. NOI and OFI processes track best bid and ask queues and change much faster than prices. They incorporate information about build-ups and depletions of order queues and they can be used to interpolate market dynamics between price changes and to track the toxicity of order flows. The proposed multiplicative model of stochastic intensities makes it possible to analyze the characteristics of the order flows as well as the instantaneous proportion of the forces of buyers and sellers without modeling the external information background. The proposed model gives the opportunity to link the micro-scale high-frequency dynamics of the limit order book with the macroscale models of stock price processes of the form of subordinated Wiener processes by means of limit theorems for special random sums and hence, to give a deeper insight in the nature of popular models of statistical regularities of the evolution of characteristics of financial markets such as generalized hyperbolic distributions and other normal variance-mean mixtures.

Summation methods applied to Voronovskaya-type theorems for the partial sums of Fourier series and for Fejér operators

Abstract We obtain Voronovskaya-type theorems for the partial sums of Fourier series using the second order Cesáro method of summation. Then we obtain two versions of Voronovskaya-type theorems for Fejér operators and finally we deduce an integral identity.

Second and third orders asymptotic expansions for the distribution of particles in a branching random walk with a random environment in time

Consider a branching random walk in which the offspring distribution and the moving law both depend on an independent and identically distributed random environment indexed by the time. For the normalised counting measure of the number of particles of generation $n$ in a given region, we give the second and third orders asymptotic expansions of the central limit theorem under rather weak assumptions on the moments of the underlying branching and moving laws. The obtained results and the developed approaches shed light on higher order expansions. In the proofs, the Edgeworth expansion of central limit theorems for sums of independent random variables, truncating arguments and martingale approximation play key roles. In particular, we introduce a new martingale, show its rate of convergence, as well as the rates of convergence of some known martingales, which are of independent interest.

A Tauberian Theorem for Double Cesàro Summability Method

We have generalized Littlewood Tauberian theorems for(C,k,r)summability of double sequences by using oscillating behavior and de la Vallée-Poussin mean. Further, the generalization of(C,r)summability from(C,k,r)summability is given as corollaries which were earlier established by the authors.

On the divergence of triangular and eccentric spherical sums of double Fourier series

We construct a continuous function on the torus with almost everywhere divergent triangular sums of double Fourier series. We also prove an analogous theorem for eccentric spherical sums. Bibliography: 14 titles.

Multidimensional limit theorems for homogeneous sums: A survey and a general transfer principle

The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant.

A Local Limit Theorem for sums of independent random vectors

We prove a local limit theorem for sums of independent random vectors satisfying appropriate tightness assumptions. In particular, the local limit theorem holds in dimension 1 if the summands are uniformly bounded.

A Probabilistic Approach to Generalized Zeckendorf Decompositions

Generalized Zeckendorf decompositions are expansions of integers as sums of elements of solutions to recurrence relations. The simplest cases are base-$b$ expansions, and the standard Zeckendorf decomposition uses the Fibonacci sequence. The expansions are finite sequences of nonnegative integer coefficients (satisfying certain technical conditions to guarantee uniqueness of the decomposition) and which can be viewed as analogues of sequences of variable-length words made from some fixed alphabet. In this paper we present a new approach and construction for uniform measures on expansions, identifying them as the distribution of a Markov chain conditioned not to hit a set. This gives a unified approach that allows us to easily recover results on the expansions from analogous results for Markov chains, and in this paper we focus on laws of large numbers, central limit theorems for sums of digits, and statements on gaps (zeros) in expansions. We expect the approach to prove useful in other similar contexts.

Building Above Read-Once Polynomials: Identity Testing and Hardness of Representation

Polynomial Identity Testing (PIT) algorithms have focussed on polynomials computed either by small alternation-depth arithmetic circuits, or by read-restricted formulas. Read-once polynomials (ROPs) are computed by read-once formulas (ROFs) and are the simplest of read-restricted polynomials. Building structures above these, we show the following: (1) a deterministic polynomial-time non-black-box PIT algorithm for $$\sum ^{(2)}\times \prod \times \mathsf{ROF}$$ź(2)ן×ROF. (2) Weak hardness of representation theorems for sums of powers of constant-free ROPs and for $$\mathsf{ROF}$$ROFs of the form $$\sum \times \prod \times \sum $$ź×ź×ź. (3) A partial characterization of multilinear monotone constant-free ROPs.

Limit theorems for weighted Bernoulli random fields under Hannan’s condition

Recently, invariance principles for partial sums of Bernoulli random fields over rectangular index sets have been proved under Hannan’s condition. In this note we complement previous results by establishing limit theorems for weighted Bernoulli random fields, including central limit theorems for partial sums over arbitrary index sets and invariance principles for Gaussian random fields. Most results improve earlier ones on Bernoulli random fields under Wu’s condition, which is stronger than Hannan’s condition.

Almost sure central limit theorem for self-normalized partial sums and maxima

Let \(X, X_1, X_2,\ldots \) be a sequence of independent and identically distributed random variables with zero mean and finite second moment. A universal result in almost sure central limit theorem for the self-normalized partial sums \(S_n/V_n\) and maxima \(M_n\) is established, where \(S_n=\sum _{i=1}^nX_i, V^2_n=\sum _{i=1}^nX^2_i,\) and \(M_n=\max _{1\le i\le n}X_i.\)

On asymmetric generalization of the Weibull distribution by scale–location mixing of normal laws

Two approaches are suggested to the definition of asymmetric generalized Weibull distribution. These approaches are based on the representation of the two-sided Weibull distributions as variance–mean normal mixtures or more general scale–location mixtures of the normal laws. Since both of these mixtures can be limit laws in limit theorems for random sums of independent random variables, these approaches can provide additional arguments in favor of asymmetric two-sided Weibull-type models of statistical regularities observed in some problems related to stopped random walks, in particular, in problems of modeling the evolution of financial markets

Asymptotic results for random sums of dependent random variables

Our main result is a central limit theorem for random sums of the form ∑i=1NnXi, where {Xi}i≥1 is a stationary m-dependent process and Nn is a random index independent of {Xi}i≥1. This extends the work of Chen and Shao on the i.i.d. case to a dependent setting and provides a variation of a recent result of Shang on m-dependent sequences. Further, a weak law of large numbers is proven for ∑i=1NnXi, and the results are exemplified with applications on moving average and descent processes.

Study on Fractal Mathematical Models of Pulverizing Theory for Ore

Fractal dimension is an important parameter to characterize size distribution and shape features of particles. However, it still remains unclear whether the fractal dimensions measured with different methods for the same particles will have the identical results, or whether the similar quantitative relationship exists between D3 and D2 or D3 and D1 as stated in the product–sum theorem. Silicon dioxide and potassium feldspar were selected for PDS measurements and 2DSPI extractions in this paper. By introducing the concept of fractal size, D3, D2 and D1 are defined in a unified way, and their relationships are obtained under the defined assumptions including the product–sum theorem. The two fractional mathematical models and the three new algorithms including the power spectrum method presented in this paper provide the solutions to the problems encountered in the measurement of fractal dimensions. The Gauss–Newton iterative method of R–R distribution is superior to the existing fitted method. The four relationships of the fractal dimensions have been verified both experimentally and by the computational models. The results show that the relationships among D3, D2 and D1 are rational and valid.

Sum and product theorems depending on the (p, q)-th order and (p, q)-th type of entire functions

The main object of the present paper is to obtain new estimates involving the (p,q)-th order and the (p,q)-th type of entire functions under some suitable conditions. Some open questions, which emerge naturally from this investigation, are also indicated as a further scope of study for the interested future researchers in this branch of Complex Analysis.