scholarly journals Convergence in probability and allied results

1969 ◽  
pp. 186-186
Author(s):  
A. R. Padmanabhan
Keyword(s):  
Convergence In Probability

scholarly journals A harmonically weighted filter for cyclical long memory processes

2021 ◽  
Author(s):  
Federico Maddanu
Keyword(s):  
Spectral Density ◽  
Long Memory ◽  
Convergence In Probability ◽  
Simulation Methods ◽  
Integrated Process ◽  
Paleoclimatic Data ◽  
Memory Time ◽  
Long Memory Processes ◽  
Short Memory ◽  
Memory Parameter

AbstractThe estimation of the long memory parameter d is a widely discussed issue in the literature. The harmonically weighted (HW) process was recently introduced for long memory time series with an unbounded spectral density at the origin. In contrast to the most famous fractionally integrated process, the HW approach does not require the estimation of the d parameter, but it may be just as able to capture long memory as the fractionally integrated model, if the sample size is not too large. Our contribution is a generalization of the HW model, denominated the Generalized harmonically weighted (GHW) process, which allows for an unbounded spectral density at $$k \ge 1$$ k ≥ 1 frequencies away from the origin. The convergence in probability of the Whittle estimator is provided for the GHW process, along with a discussion on simulation methods. Fit and forecast performances are evaluated via an empirical application on paleoclimatic data. Our main conclusion is that the above generalization is able to model long memory, as well as its classical competitor, the fractionally differenced Gegenbauer process, does. In addition, the GHW process does not require the estimation of the memory parameter, simplifying the issue of how to disentangle long memory from a (moderately persistent) short memory component. This leads to a clear advantage of our formulation over the fractional long memory approach.

scholarly journals Rate of convergence in probability to the Marchenko-Pastur law

Bernoulli ◽  
2004 ◽  
Vol 10 (3) ◽  
pp. 503-548 ◽  
Author(s):  
Friedrich Götze ◽  
Alexander Tikhomirov
Keyword(s):  
Rate Of Convergence ◽  
Convergence In Probability

Characterization of stable processes by identically distributed stochastic integrals

1980 ◽  
Vol 12 (3) ◽  
pp. 689-709 ◽  
Author(s):  
M. Riedel
Keyword(s):  
Stochastic Process ◽  
Stable Process ◽  
Convergence In Probability ◽  
Stochastic Integrals ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Subject

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

scholarly journals Tree Polymers in the Infinite Volume Limit at Critical Strong Disorder

2011 ◽  
Vol 48 (03) ◽  
pp. 885-891
Author(s):  
Torrey Johnson ◽  
Edward C. Waymire
Keyword(s):  
Asymptotic Variance ◽  
Standard Theory ◽  
Convergence In Probability ◽  
Infinite Volume ◽  
Weak Disorder ◽  
Branching Random Walks ◽  
Strong Disorder ◽  
Volume Limit ◽  
Specific Formula ◽  
Infinite Volume Limit

The almost-sure existence of a polymer probability in the infinite volume limit is readily obtained under general conditions of weak disorder from standard theory on multiplicative cascades or branching random walks. However, speculations in the case of strong disorder have been mixed. In this note existence of an infinite volume probability is established at critical strong disorder for which one has convergence in probability. Some calculations in support of a specific formula for the almost-sure asymptotic variance of the polymer path under strong disorder are also provided.

Convergence in Lp Norm

2021 ◽  
pp. 418-437
Author(s):  
James Davidson
Keyword(s):  
Law Of Large Numbers ◽  
Almost Sure Convergence ◽  
Convergence In Probability ◽  
Martingale Differences ◽  
Lp Norm ◽  
Large Numbers

This chapter looks in detail at proofs of the weak law of large numbers (convergence in probability) using the technique of establishing convergence in Lp‐norm. The extension to a proof of almost‐sure convergence is given, and then special results for martingale differences, mixingales, and approximable processes. These results are proved in array notation to allow general forms of heterogeneity.

scholarly journals Gaps in Discrete Random Samples

2009 ◽  
Vol 46 (04) ◽  
pp. 1038-1051 ◽  
Author(s):  
Rudolf Grübel ◽  
Paweł Hitczenko
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Borderline Case ◽  
Convergence In Probability ◽  
Enumerative Combinatorics ◽  
Random Samples ◽  
The Family ◽  
Heavy Tailed ◽  
Convergence Almost Surely ◽  
Necessary And Sufficient

Let (X i ) i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ0 of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X 1,…,X n } of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (q k ) k∈ℕ0 , q k =P(X 1≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑ k=0 ∞ q k+1/q k <∞ and limk→∞ q k+1/q k =0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if q k+1/q k → 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme–Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.

Twister Generator of Poisson Random Numbers with the Use of Cumulative Frequency Technology

2020 ◽  
pp. 101-123
Author(s):  
A.F. Deon ◽  
D.D. Dmitriev ◽  
Yu.A. Menyaev
Keyword(s):  
Poisson Distribution ◽  
Probability Distributions ◽  
Random Variables ◽  
Convergence In Probability ◽  
Random Numbers ◽  
Cumulative Frequency ◽  
New Approach ◽  
Frequency Distributions ◽  
Simulation Results

The widely known generators of Poisson random variables are associated with different modifications of the algorithm based on the convergence in probability of a sequence of uniform random variables to the created stochastic number. However, in some situations, this approach yields different discrete Poisson probability distributions and skipping in the generated numbers. This paper offers a new approach for creating Poisson random variables based on the complete twister generator of uniform random variables, using cumulative frequency technology. The simulation results confirm that probabilistic and frequency distributions of the obtained stochastic numbers completely coincide with the theoretical Poisson distribution. Moreover, combining this new approach with the tuning algorithm of basic twister generation allows for a significant increase in length of the created sequences without using additional RAM of the computer

Convergence Results for r-Iterated Means of the Denominators of the Lüroth Series

2016 ◽  
Vol 11 (2) ◽  
pp. 179-203
Author(s):  
Rita Giuliano
Keyword(s):  
Convergence In Probability ◽  
Convergence In Distribution ◽  
Asymptotic Results ◽  
Arithmetic Means ◽  
Convergence Results ◽  
Weighted Means

Abstract In the present paper we extend two classic asymptotic results concerning convergence in probability and convergence in distribution for the denominators of the Lüroth series and obtain new theorems concerning the same two kinds of convergence for the r-iterated arithmetic means of such denominators. These results are extended to r-iterated weighted means.

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