Sequence Of Random Elements Research Articles

Product disintegrations: A law of large numbers via conditional independence

We introduce the concept of a product disintegration, which generalizes exchangeability by allowing one to render conditional independence in terms of a suitable hidden sequence of random probabilities. We prove that, given a sequence of random elements in a compact metric space, a strong law of large numbers (SLLN) holds for observables of this process if and only if the process admits a product disintegration that itself satisfies a type of SLLN. As an application, we provide a characterization of the SLLN for identically distributed (not necessarily independent) Bernoulli sequences.

ON THE ACCURACY BE THE METRIC OF THE DENSITY ESTIMATION CONSTRUCTED BY DEPENDENT OBSERVATIONS

ON THE ACCURACY BE THE METRIC OF THE DENSITY ESTIMATION CONSTRUCTED BY DEPENDENT OBSERVATIONS

Some Strong Laws for Normed Weighted Sums of Stochastically Dominated Banach Space Valued Random Elements Irrespective of Their Joint Distributions

Let {Vn, n ≥ 1} be a sequence of random elements in a real separable Banach space and suppose that {Vn, n ≥ 1} is stochastically dominated by a random element V. Let {an, n ≥ 1} and {bn, n ≥ 1} be real sequences with 0 < bn ↑ ∞. Conditions are provided under which {anVn, n ≥ 1} obeys the general strong law of large numbers almost surely irrespective of the joint distributions of the {Vn, n ≥ 1}. No geometric conditions are imposed on the underlying Banach space. Examples are provided which illustrate, compare, or demonstrate the sharpness of the results.

Some strong laws of large numbers for blockwise martingale difference sequences in martingale type p Banach spaces

For a blockwise martingale difference sequence of random elements {V n , n ≥ 1} taking values in a real separable martingale type p (1 ≤ p ≤ 2) Banach space, conditions are provided for strong laws of large numbers of the form $$\lim _{n \to \infty } \sum\nolimits_{i = 1}^n {V_i /g_n = 0}$$ almost surely to hold where the constants g n ↑ ∞. A result of Hall and Heyde [Martingale Limit Theory and Its Application, Academic Press, New York, 1980, p. 36] which was obtained for sequences of random variables is extended to a martingale type p (1 < p ≤ 2) Banach space setting and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.

Almost sure functional central limit theorems for weakly dependent sequences

In this paper, we give a general almost sure functional central limit theorem for a sequence of random elements. We apply the result to various kinds of weakly dependent sequences.

On Some Properties of Randomly Indexed Sequences of Random Elements

Let \((\Omega ,\mathcal{A},P)\) be a probability space with a nonatomic measure P and let (S,ρ) be a separable complete metric space. Let {Nn,n≥1} be an arbitrary sequence of positive-integer valued random variables. Let {Fk,k≥1} be a family of probability laws and let X be some random element defined on \((\Omega ,\mathcal{A},P)\) and taking values in (S,ρ).

On Complete Convergence in Mean of Normed Sums of Independent Random Elements in Banach Spaces

For a sequence of random elements {T n , n ≥ 1} in a real separable Banach space 𝒳, we study the notion of T n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T n converging completely to 0 and (ii) T n converging to 0 in mean of order p. When 𝒳 is of Rademacher type p (1 ≤ p ≤ 2), for a sequence of independent mean 0 random elements {V n , n ≥ 1} in 𝒳 and a sequence of constants b n → ∞, conditions are provided under which the normed sum converges completely to 0 in mean of order p. Moreover, these conditions for converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.

Integral analogues of almost sure limit theorems

An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable processes, moreover for processes being in the domain of attraction of a stable law or being in the domain of geometric partial attraction of a semistable or a max-semistable law.

Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces

Let (Xi)i≥1 be an i.i.d. sequence of random elements in the Banach space B, Sn≔X1+⋅⋅⋅+Xn and ξn be the random polygonal line with vertices (k/n,Sk), k=0,1,...,n. Put ρ(h)=hαL(1/h), 0≤h≤1 with 0 t)=o(t−p(α)) where p(α)=1/(1/2−α). This completes Lamperti (1962) invariance principle.

Some Strong Laws of Large Numbers for Banach Space Valued Summands Irrespective of Their Joint Distributions

Conditions are provided under which a sequence of random elements in a real separable Banach space obeys a general strong law of large numbers. No conditions are imposed on the joint distributions of the random elements or on the Banach space, and the results are new even when the Banach space is the real line. Illustrative examples are provided which compare the results or which show that they are sharp.

Characterization of Strong Limits of Randomly Indexed Sequences

Let fXn ;n >1g be an arbitrary sequence of random elements dened on a probability space (;A;P) with a nonatomic measure P and taking values in a separable complete metric space (S;). In this paper we characterize the set of all possible weak limits of the sequences fXNn ;n >1g, where fNn ;n >1g is a sequence of positive integer-valued random variables. The proof shows how, for a given probability law F(), we can dene a random sequence fNn ;n >1g satisfying XNn D !F(). c 2000 Elsevier Science B.V. All rights reserved MSC: 60F05

A general non-convex large deviation result with applications to stochastic equations

We prove an abstract large deviation result for a sequence of random elements of a vector space satisfying an “abstract exponential martingale condition”. The framework naturally generates non-convex rate functions. We apply the result to solutions of Ito stochastic equations in R d driven by Brownian motion and a Poisson random measure.

On convergence of series of independent random elements in banach spaces

The rate of convergence for an almost certainly convergent series independent random elements in a real separable Banach space is studied in this paper. More specifically, when S n converges almost certainly to a random element S, the tail series is a well defined sequence of random elements with almost certainly. The main result establishes for a sequence of positive constants with b j ⩽ Const. b n whenever the equivalence between the tail series weak law of large numbers and the limit law thereby extending a result of Nam and Rosalsky [20] to a Banach space setting while also simplifying the argument used in the earlier result. The quasimonotonicity proviso on cannot be dispensed with

On the rate of convergence in the central limit theorem for sequences of weakly dependent Hilbert-valued random variables

A. N. Tikhomirov UDC 519.2 Let X1, X2,... be a sequence of random elements of the Hilbert space H with zero mean, EXj = O, and covariation operators Aj = cov(Xj). Suppose the sequence { Xj : j = 1... } satisfies the uniformly strong mixing condition with coefficient ~(m). Let

Coupling and convergence of random elements, processes, and regenerative processes

The paper starts by proving that a sequence of random elements can be coupled in such a way that the random elements eventually coincide if and only if liminf of their densities is a density. It continues with a survey of some general coupling theory for stochastic processes and applications to wide sense regenerative processes and Palm theory. Finally, a successful coupling and ɛ-coupling of wide sense regenerative processes is constructed without assuming that the inter-regeneration times have finite mean.

Synthesis and recognition of sequences

A string or sequence is a linear array of symbols that come from an alphabet. Due to unknown substitutions, insertions, and deletions of symbols, a sequence cannot be treated like a vector or a tuple of a fixed number of variables. The synthesis of an ensemble of sequences is a sequence of random elements that specify the probabilities of occurrence of the different symbols at the corresponding sites of the sequences. The synthesis is determined by a hierarchical sequence synthesis procedure (HSSP), which returns not only the taxonomic hierarchy of the whole ensemble of sequences but also the alignment and the synthesis of a group (a subset of the ensemble) of the sequences at each level of the hierarchy. The HSSP does not require the ensemble of sequences to be presented in the form of a tabulated array of data, the hierarchical information of the data, or the assumption of a stochastic process. The authors present the concept of sequence synthesis and the applicability of the HSSP as a supervised classification procedure as well as an unsupervised classification procedure.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Limit theorems for randomly weighted sums of random elements in normed linear spaces

Let { V n ; n ≥ 1} be a sequence of random elements in a separable normed linear space E, uniformly dominated by a random variable V. Let { A nk ; k = 1, 2, …, n; n ≥ 1} be a triangular array of random variables. In this paper, conditions for the convergence of Σ k=1 n A nk V k to zero (in probability and completely) are obtained. No geometric condition on E is imposed.

Stationarity in models of successive stochastic phenomena

It is well known that the image of a stationary stochastic process under a deterministic, measurable function that ignores the past is again a stationary stochastic process. This result allows us to assert the stationarity of a model for a phenomenon that is made up of a random component that is stationary, followed by a deterministic operation. Many models of real stochastic phenomena, however, postulate two or more random mechanisms acting successively. To investigate the stationarity of these models, we view the second random mechanism as a random transformation of the first one, and ask for conditions under which stationarity is preserved by the action of a single randomly chosen transformation, or a stationary sequence of random transformations. Notions of stationarity and ergodicity for sequences of random elements of a function space are used to find various conditions under which stationarity, or stationarity and ergodicity, are preserved by certain deterministic and random transformations, and to treat some examples from applications.

Weak convergence of sequences of random elements with random indices

Let (S, d) be a separable metric space equipped with its Borel σ field . Let {Yn, n ≥ 1} be a sequence of S-valued random elements defined on a probability space (Ω, , p). Assume Yn ⇒ Y converges weakly to an S-valued random element Y. Let {Nn, n ≥ 1} be a sequence of positive integer-valued random variables defined on the same probability space (Ω, , p).

Limit theorems for weighted sums of random elements in separable Banach spaces

Let a nk , n ≥ 1, k ≥ 1, be a double array of real numbers and let V n , n ≥ 1, be a sequence of random elements taking values in a separable Banach space. In this paper, we examine under what conditions the sequence Σ k≥1 a nk V k , n ≥ 1, has a limit either in probability or almost surely.