Some limit theorems of runs to the continuous‐valued sequence

Kybernetes ◽  
2008 ◽  
Vol 37 (9/10) ◽  
pp. 1279-1286 ◽  
Author(s):  
Fan Aihua ◽  
Wang Zhongzhi ◽  
Ding Fangqing
Keyword(s):  
Limit Theorems ◽  
Random Sequence ◽  
Random Variables ◽  
Moment Generating Function ◽  
Upper And Lower Bounds ◽  
Content Type ◽  
Independent Case ◽  
The Moment ◽  
Martingale Convergence ◽  
Crucial Part

PurposeThe purpose of this paper is to give some limit theorems on ε‐neighborhood and ε‐increasing runs of continuous‐valued dependent random sequence. In the main result no assumptions are made concerning the random variables. As corollary a result on independent case is obtained.Design/methodology/approachThe crucial part of the proof is to construct a non‐negative supper‐martingale depending on a parameter by using the moment generating function, and then applying the Doob's martingale convergence theorem.FindingsThe upper and lower bounds of the deviations from the sums of arbitrary continuous‐valued random variables from the reference distributions are obtained.Research limitations/implicationsThe computation of asymptotic log‐likelihood ratio h(P|Q) is the main limitations, and it is difficult to obtain the rigorous bounds of the deviations.Practical implicationsA useful method to study the property for runs of dependent random sequence.Originality/valueThe new approach of study strong limit behavior for dependent random sequence.

scholarly journals Almost sure central limit theorems for m-dependent random variables

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5581-5590 ◽  
Author(s):  
Yu Miao ◽  
Xiaoyan Xu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Sequence ◽  
Random Variables ◽  
Dependent Random Variables ◽  
Moment Conditions ◽  
Dependent Sequence ◽  
The Moment

In the present paper, the almost sure central limit theorem for them-dependent random sequence is established, which weakens the moment conditions of Giuliano [10] for the stationary m-dependent sequence and gets the same results with different methods.

Functional limit theorems for high-level subcritical branching processes in random environment

2014 ◽  
Vol 24 (5) ◽  
Author(s):  
Valeriy I. Afanasyev
Keyword(s):  
Random Environment ◽  
Limit Theorems ◽  
Branching Process ◽  
Branching Processes ◽  
Moment Generating Function ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
The Moment ◽  
High Level ◽  
Passage Times

AbstractThe paper is concerned with subcritical branching process in random environment. It is assumed that the moment-generating function of steps of the associated random walk is equal to 1 for some positive value of the argument. Functional limit theorems for sizes of various generations and passage times to various levels are put forward.

scholarly journals Outage Probability and Normalized SINR-Based Power Allocation over Rician Fading Channels

2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Danh H. Ho ◽  
T. Aaron Gulliver
Keyword(s):  
Energy Efficiency ◽  
Power Consumption ◽  
Outage Probability ◽  
Power Allocation ◽  
Fading Channels ◽  
Moment Generating Function ◽  
Upper And Lower Bounds ◽  
Rician Fading ◽  
The Moment ◽  
Rician Fading Channels

This paper considers power allocation in cellular networks over Rician fading channels. The goal is to improve the power consumption and energy efficiency as well as satisfy as many users as possible subject to user outage probability and normalized signal to interference plus noise ratio (SINR) constraints. The exact outage probability over Rician fading channels is determined using the moment-generating function (MGF). Further, upper and lower bounds on the outage probability are derived. These are used to characterize the relationship between outage probability and normalized SINR in Rician fading channels. Power allocation algorithms for power minimization and energy efficiency are proposed. Simulation results are presented to compare the performance of the proposed schemes with existing methods in terms of power consumption, throughput, energy efficiency, outage probability, and number of unsatisfied users.

scholarly journals Efficient importance sampling for large sums of independent and identically distributed random variables

2021 ◽  
Vol 31 (6) ◽  
Author(s):  
Nadhir Ben Rached ◽  
Abdul-Lateef Haji-Ali ◽  
Gerardo Rubino ◽  
Raúl Tempone
Keyword(s):  
Importance Sampling ◽  
Rare Event ◽  
Random Variables ◽  
Moment Generating Function ◽  
Accurate Estimate ◽  
Probability Density Functions ◽  
Density Functions ◽  
The Moment ◽  
Log Normal ◽  
Independent And Identically Distributed

AbstractWe discuss estimating the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $$\mathbb {P}(\sum _{i=1}^{N}{X_i} \le \gamma )$$ P ( ∑ i = 1 N X i ≤ γ ) , via importance sampling (IS). We are particularly interested in the rare event regime when N is large and/or $$\gamma $$ γ is small. The exponential twisting is a popular technique for similar problems that, in most cases, compares favorably to other estimators. However, it has some limitations: (i) It assumes the knowledge of the moment-generating function of $$X_i$$ X i and (ii) sampling under the new IS PDF is not straightforward and might be expensive. The aim of this work is to propose an alternative IS PDF that approximately yields, for certain classes of distributions and in the rare event regime, at least the same performance as the exponential twisting technique and, at the same time, does not introduce serious limitations. The first class includes distributions whose probability density functions (PDFs) are asymptotically equivalent, as $$x \rightarrow 0$$ x → 0 , to $$bx^{p}$$ b x p , for $$p>-1$$ p > - 1 and $$b>0$$ b > 0 . For this class of distributions, the Gamma IS PDF with appropriately chosen parameters retrieves approximately, in the rare event regime corresponding to small values of $$\gamma $$ γ and/or large values of N, the same performance of the estimator based on the use of the exponential twisting technique. In the second class, we consider the Log-normal setting, whose PDF at zero vanishes faster than any polynomial, and we show numerically that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting IS PDF. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large N and/or small $$\gamma $$ γ .

ALMOST SURE CENTRAL LIMIT THEOREMS OF THE PARTIAL SUMS AND MAXIMA FROM COMPLETE AND INCOMPLETE SAMPLES OF STATIONARY SEQUENCES

2012 ◽  
Vol 12 (03) ◽  
pp. 1150026 ◽  
Author(s):  
ZUOXIANG PENG ◽  
BIN TONG ◽  
SARALEES NADARAJAH
Keyword(s):  
Central Limit ◽  
Random Vector ◽  
Limit Theorems ◽  
Random Sequence ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Partial Sums ◽  
Stationary Sequences ◽  
Vector Formula ◽  
Weakly Dependent

Let (Xn) denote an independent and identically distributed random sequence. Let [Formula: see text] and Mn = max {X1, …, Xn} be its partial sum and maximum. Suppose that some of the random variables of X1, X2,… can be observed and denote by [Formula: see text] the maximum of observed random variables from the set {X1, …, Xn}. In this paper, we consider the joint limiting distribution of [Formula: see text] and the almost sure central limit theorems related to the random vector [Formula: see text]. Furthermore, we extend related results to weakly dependent stationary Gaussian sequences.

scholarly journals Tail Bounds for the Wiener Index of Random Trees

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Tämur Ali Khan ◽  
Ralph Neininger
Keyword(s):  
Lower Bounds ◽  
Random Search ◽  
Wiener Index ◽  
Moment Generating Function ◽  
Random Trees ◽  
Upper And Lower Bounds ◽  
Search Trees ◽  
Tail Probabilities ◽  
International Audience ◽  
The Moment

International audience Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.

scholarly journals On measure concentration in graph products

2009 ◽  
Vol 50 ◽  
Author(s):  
Matas Šileikis
Keyword(s):  
Generating Function ◽  
Random Variables ◽  
Moment Generating Function ◽  
Graph Products ◽  
Grid Graph ◽  
Connected Graphs ◽  
Measure Concentration ◽  
Product Graph ◽  
The Moment ◽  
The One

Bollobás and Leader [1] showed that among the n-fold products of connected graphs of order k the one with minimal t-boundary is the grid graph. Given any product graph G and a set A of its vertices that contains at least half of V (G), the number of vertices at a distance at least t from A decays (as t grows) at a rate dominated by P(X1 + . . . + Xn  \geq   t) where Xi are some simple i.i.d. random variables. Bollobás and Leader used the moment generating function to get an exponentialbound for this probability. We insert a missing factor in the estimate by using a somewhat more subtle technique (cf. [3]).

Welcome to the marriage of human capability and artificial intelligence

2019 ◽  
Vol 19 (1) ◽  
pp. 10-14
Author(s):  
Ryan Scott ◽  
Malcolm Le Lievre
Keyword(s):  
Artificial Intelligence ◽  
Team Building ◽  
Workplace Change ◽  
Content Type ◽  
The Future ◽  
Human Capability ◽  
Mind Set ◽  
Future Work ◽  
Crucial Part ◽  
Practical Implications

Purpose The purpose of this paper is to explore insights methodology and technology by using behavioral to create a mind-set change in the way people work, especially in the age of artificial intelligence (AI). Design/methodology/approach The approach is to examine how AI is driving workplace change, introduce the idea that most organizations have untapped analytics, add the idea of what we know future work will look like and look at how greater, data-driven human behavioral insights will help prepare future human-to-human work and inform people’s work with and alongside AI. Findings Human (behavioral) intelligence will be an increasingly crucial part of behaviorally smart organizations, from hiring to placement to adaptation to team building, compliance and more. These human capability insights will, among other things, better prepare people and organizations for changing work roles, including working with and alongside AI and similar tech innovation. Research limitations/implications No doubt researchers across the private, public and nonprofit sectors will want to further study the nexus of human capability, behavioral insights technology and AI, but it is clear that such work is already underway and can prove even more valuable if adopted on a broader, deeper level. Practical implications Much “people data” inside organizations is currently not being harvested. Validated, scalable processes exist to mine that data and leverage it to help organizations of all types and sizes be ready for the future, particularly in regard to the marriage of human capability and AI. Social implications In terms of human capability and AI, individuals, teams, organizations, customers and other stakeholders will all benefit. The investment of time and other resources is minimal, but must include C-suite buy in. Originality/value Much exists on the softer aspects of the marriage of human capability and AI and other workplace advancements. What has been lacking – until now – is a 1) practical, 2) validated and 3) scalable behavioral insights tech form that quantifiably informs how people and AI will work in the future, especially side by side.

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