Limit theorems for cumulative processes

Introduction and summary. From the naive viewpoint of a nonprobabilist, the remarkable fact about the central limit theorem is that (under suitable conditions) the average of a large number of random variables converges (in a suitable sense) to a limit which is independenit of the things being averaged, but is completely determined by the averaging process itself. In the present paper, this situation is abstracted from its connection with probability. As we shall see, the restriction to positive functionals which characterizes probability is in no way necessary for validity of the limit theorems. Moreover, we are able to dispense completely with measure theory, in both definitions and proofs. In a purely algebraic and function-space context, we find a whole sequence of averaging processes, each with its own limit function; the first two of them generalize the weak law of large numbers and the central limit theorem. We prove convergence in three different senses, under three different sets of assumptions: first convergence of moments, then uniform convergence of characteristic functions, and finally weak convergence of distribution functions. Our first theorem is purely combinatorial; our next, essentially a calculus theorem; in our third we use function-space techniques borrowed from the modern theory of partial differential equations. By doing so, we obtain rather refined and precise results on the strongest norm in which we can assert convergence. In this respect our results may be of interest even in the classical probabilistic cases. In particular (see Corollary 2 below) we show that if a very mild extra regularity condition is imposed on the summands in the classical central limit theorem, then not only do the partial sums converge in law to the Gaussian, but every derivative of the c.d.f. of the partial sum exists and converges to the corresponding derivative of the Gaussian. Theorem 3 possesses an interpretation in terms of electrical engineering, which is given in the Remark following Corollary 2. It also has a surprising application to numerical analysis, which is described in the Postscript. We begin with an abstract algebra A over the real numbers, equipped with a linear functional Y. (If a E A were a random variable, Y(a`) would be its kth

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

On the Chebyshev-Cramér Asymptotic Expansions

In Chapter 3 we studied the conditions under which the distributions of sums of independent random variables will converge to a limit distribution. We were interested in the weak convergence of distributions; the corresponding limit theorems are called the weak limit theorems, contrary to the strong limit theorems making use of the convergence almost surely. The necessary and sufficient conditions were found for the weak convergence of distributions of sums of independent random variables to a given infinitely divisible distribution. These conditions are quite complicated but are very helpful when we are looking for simpler conditions of convergence to a prescribed infinitely divisible distribution such as a normal, a degenerate, or a Poisson distribution.

Introduction. This is a sequel to my paper [1]. The present developments are largely independent of the previous results except in so far as given in the Appendix. Theorem 1 shows a kind of solidarity among the states of a recurrent class; it generalizes a classical result due to Kolmogorov and permits a classification of recurrent states and classes. In ?2 some relations involving the mean recurrence and first passage times are given. In ??3-5 sequences of random variables associated in a natural way with a Markov chain are studied. Theorem 2 is a generalized ergodic theorem which applies to any recurrent class, positive or null. It turns out that in a null class there is a set of numbers which plays the role of stationary absolute probabilities. In the case of a recurrent random walk with independent, stationary steps these numbers are all equal to one and the result is particularly simple. Theorem 3 shows that the kind of solidarity exhibited in Theorem 1 persists in such a sequence; it leads to the clarification of certain conditions stated by Doblin(2) in connection with his central limit theorem. Using a fundamental idea due to Doblin, the weak and strong laws of large numbers, the central limit theorem, the law of the iterated logarithm, and the limit theorems for the maxima of the associated sequence are proved very simply. Owing to the great simplicity of the method it is the conditions of validity of these limit theorems that should deserve attention. Among other things, we shall show by an example that a certain set of conditions, attributed to Kolmogorov, is in reality not sufficient for the validity of the central limit theorem. Furthermore, conditions of validity for the strong limit theorems and the limit theorems for the maxima are obtained by a rather natural strengthening of corresponding conditions for the weak limit theorems. A word about the connection of these conditions with martingale theory closes the paper. 1. The sequence of random variables {Xn}, n=0, 1, 2, * , forms a denumerable Markov chain with stationary transition probabilities. The states will be denoted by the non-negative integers(3) 0, 1, 2, * * . The n-step transition probability from the state i to the state j will be denoted by P(n) (P(l) = Pij). Thus we have

Cycle symmetry, limit theorems, and fluctuation theorems for diffusion processes on the circle

The limit theorems for a previous k-sum dependent model

A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

Inspired by service systems such as telephone call centers, we develop limit theorems for a large class of stochastic service network models. They are a special family of nonstationary Markov processes where parameters like arrival and service rates, routing topologies for the network, and the number of servers at a given node are all functions of time as well as the current state of the system. Included in our modeling framework are networks of M_t/M_t/n_t queues with abandonment and retrials. The asymptotic limiting regime that we explore for these networks has a natural interpretation of scaling up the number of servers in response to a similar scaling up of the arrival rate for the customers. The individual service rates, however, are not scaled. We employ the theory of strong approximations to obtain functional strong laws of large numbers and functional central limit theorems for these networks. This gives us a tractable set of network fluid and diffusion approximations. A common theme for service network models with features like many servers, priorities, or abandonment is “non-smooth” state dependence that has not been covered systematically by previous work. We prove our central limit theorems in the presence of this non-smoothness by using a new notion of derivative.

This study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process.

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

In this paper, we consider some functionals of the sums of independent identically distributed random variables. The functionals of the sums are important in probabilistic models and stochastic branching systems. In connection with the application in various probabilistic models and stochastic branching systems, we are interested in the fulfillment of the law of large numbers and the Central limit theorem for these sums. The main hypotheses of the paper are the presence of second order moments of the variables and the fulfillment of the Lindeberg condition is considered. The research object and subject of this paper consists of specially generated random variables using the sums of non-bound random variables. In total, 6 different sums in a special form were studied in the paper and this sum was not previously studied by other scientists. The purpose of the paper is to examine whether these sums in a special form satisfy the terms of the law of large numbers and the Central limit theorem. The main result of the paper is to show that the law of large numbers and the terms of the classical limit theorem are fulfilled in some cases. The results obtained in the paper are of theoretical importance, The Central limit theorem analogues proved here are applications of Lindeberg theorem. The results can be applied to the determination of the fluctuation of immigration branching systems as well as the asymptotic state of autoregression processes. At the same time, from the main results obtained in the paper it can be used in practical lessons conducted on the theory of probability. The results of the paper will be an important guide for young researchers. Important theorems proved in the paper can be used in probability theory, stochastic branching systems and other practical problems.

SYNOPTIC ABSTRACTIn this paper, a new class of dependent Bernoulli random variables is defined. Here the probability of success at a given trial is a function of the number of successes and probabilities of successes in the previous trials. The moment structure for this model is derived. Further, the strong law of large numbers, the central limit theorem and the law of iterated logarithm are established under a condition that the success probabilities be monotone. Simulations are carried out to demonstrate the law of large numbers and the central limit theorem.

It is well known that the sample mean, based on a sequence of independent random variables with common distribution, is a weakly consistent estimator as well as a strongly consistent estimator for the population mean. The first property is the famous weak law of large numbers (Khinchin), the second property is the strong law of large numbers (Kolmogorov). Both laws are key results of modern probability theory. Similar properties hold for more complicated sequences of random variables, for example, for sequences satisfying the stationarity assumption or for martingale sequences. These extensions are of crucial importance for statistical applications in ergodic theory and survival analysis.

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.