Stochastic Processes and their Applications, ISSN 0304-4149, 12/2016, Volume 126, Issue 12, pp. 3854 - 3864

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f(t0)>0, f′(t0)<0, and f...

Strassen | Grenander | Monotone density | Limsup | Liminf | Law of iterated logarithm | Limit set | Switching | Strong invariance theorem | Local empirical process | FUNCTIONAL LAWS | BROWNIAN-MOTION | STATISTICS & PROBABILITY | DENSITY | switching | law of iterated logarithm | local empirical process | strong invariance theorem | liminf | limsup | monotone density | limit set

Strassen | Grenander | Monotone density | Limsup | Liminf | Law of iterated logarithm | Limit set | Switching | Strong invariance theorem | Local empirical process | FUNCTIONAL LAWS | BROWNIAN-MOTION | STATISTICS & PROBABILITY | DENSITY | switching | law of iterated logarithm | local empirical process | strong invariance theorem | liminf | limsup | monotone density | limit set

Journal Article

Acta mathematica Hungarica, ISSN 1588-2632, 2007, Volume 118, Issue 1-2, pp. 155 - 170

It is proved that two types of discrepancies of the sequence {θ n x} obey the law of the iterated logarithm with the same constant...

primary 10K30 | discrepancies | law of the iterated logarithm | lacunary series | secondary 10K05 | 42A55 | Mathematics, general | Mathematics | 60F15 | Lacunary series | Discrepancies | Law of the iterated logarithm | MATHEMATICS | RIESZ-RAIKOV SUMS | SIGMA-F(NKX) | CENTRAL-LIMIT-THEOREM | ASYMPTOTIC-BEHAVIOR

primary 10K30 | discrepancies | law of the iterated logarithm | lacunary series | secondary 10K05 | 42A55 | Mathematics, general | Mathematics | 60F15 | Lacunary series | Discrepancies | Law of the iterated logarithm | MATHEMATICS | RIESZ-RAIKOV SUMS | SIGMA-F(NKX) | CENTRAL-LIMIT-THEOREM | ASYMPTOTIC-BEHAVIOR

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 6/2019, Volume 32, Issue 2, pp. 721 - 736

In this paper, we present Chung’s functional law of the iterated logarithm for increments of a fractional Brownian motion...

60G22 | Chung’s functional law of the iterated logarithm | Increments | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Fractional Brownian motion | 60F17 | 60F15 | STATISTICS & PROBABILITY | Chung's functional law of the iterated logarithm

60G22 | Chung’s functional law of the iterated logarithm | Increments | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Fractional Brownian motion | 60F17 | 60F15 | STATISTICS & PROBABILITY | Chung's functional law of the iterated logarithm

Journal Article

Bernoulli, ISSN 1350-7265, 11/2017, Volume 23, Issue 4A, pp. 2330 - 2379

Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs...

Stable-like process | Sample path | Symmetric jump processes | Range | Local time | Law of the iterated logarithm | BROWNIAN-MOTION | DIFFUSION-PROCESSES | LEVY PROCESSES | symmetric jump processes | range | STATISTICS & PROBABILITY | stable-like process | SAMPLE PATH PROPERTIES | MARKOV-PROCESSES | LOCAL-TIMES | D-SETS | UPPER-BOUNDS | law of the iterated logarithm | sample path | PARABOLIC HARNACK INEQUALITY | METRIC MEASURE-SPACES | local time

Stable-like process | Sample path | Symmetric jump processes | Range | Local time | Law of the iterated logarithm | BROWNIAN-MOTION | DIFFUSION-PROCESSES | LEVY PROCESSES | symmetric jump processes | range | STATISTICS & PROBABILITY | stable-like process | SAMPLE PATH PROPERTIES | MARKOV-PROCESSES | LOCAL-TIMES | D-SETS | UPPER-BOUNDS | law of the iterated logarithm | sample path | PARABOLIC HARNACK INEQUALITY | METRIC MEASURE-SPACES | local time

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2016, Volume 59, Issue 12, pp. 2503 - 2526

Kolmogorov＇s exponential inequalities are basic tools for studying the strong limit theorems such as the classical laws of the iterated logarithm for both independent and dependent random variables...

Kolmogorov’s exponential inequality | central limit theorem | negative dependence | sub-linear expectation | Mathematics | Applications of Mathematics | laws of the iterated logarithm | 60F05 | capacity | 60F15 | MATHEMATICS, APPLIED | NONADDITIVE PROBABILITIES | SEQUENCES | THEOREM | Kolmogorov's exponential inequality | RANDOM-VARIABLES | MATHEMATICS | MOMENT INEQUALITIES | WEAK-CONVERGENCE | FRAMEWORK | Lower bounds | Theorems | Mathematical analysis | Inequalities | Random variables | Logarithms | Sums | Mathematics - Probability

Kolmogorov’s exponential inequality | central limit theorem | negative dependence | sub-linear expectation | Mathematics | Applications of Mathematics | laws of the iterated logarithm | 60F05 | capacity | 60F15 | MATHEMATICS, APPLIED | NONADDITIVE PROBABILITIES | SEQUENCES | THEOREM | Kolmogorov's exponential inequality | RANDOM-VARIABLES | MATHEMATICS | MOMENT INEQUALITIES | WEAK-CONVERGENCE | FRAMEWORK | Lower bounds | Theorems | Mathematical analysis | Inequalities | Random variables | Logarithms | Sums | Mathematics - Probability

Journal Article

Journal of inequalities and applications, ISSN 1029-242X, 2018, Volume 2018, Issue 1, pp. 1 - 17

.... In this paper, by the Kolmogorov type maximal inequality and Stein’s method, we establish the result of the law of the iterated logarithm for LNQD sequence with less restriction of moment conditions...

Stein’s method | law of the iterated logarithm | Analysis | Mathematics, general | Beveridge and Nelson decomposition | Mathematics | linear process | Applications of Mathematics | LNQD sequence | COMPLETE CONVERGENCE | STATIONARY LINEAR-PROCESSES | MATHEMATICS, APPLIED | INEQUALITIES | DEPENDENT RANDOM-VARIABLES | WEIGHTED SUMS | MATHEMATICS | PRODUCTS | Stein's method | ASSOCIATION | CENTRAL-LIMIT-THEOREM | Theorems | Queuing theory | Random variables | Research

Stein’s method | law of the iterated logarithm | Analysis | Mathematics, general | Beveridge and Nelson decomposition | Mathematics | linear process | Applications of Mathematics | LNQD sequence | COMPLETE CONVERGENCE | STATIONARY LINEAR-PROCESSES | MATHEMATICS, APPLIED | INEQUALITIES | DEPENDENT RANDOM-VARIABLES | WEIGHTED SUMS | MATHEMATICS | PRODUCTS | Stein's method | ASSOCIATION | CENTRAL-LIMIT-THEOREM | Theorems | Queuing theory | Random variables | Research

Journal Article

Annals of Operations Research, ISSN 0254-5330, 5/2018, Volume 264, Issue 1, pp. 157 - 191

A functional law of the iterated logarithm (FLIL) and its corresponding law of the iterated logarithm (LIL...

Strong approximation | Customer feedback | Business and Management | Functional law of the iterated logarithm | Operations Research/Decision Theory | Theory of Computation | Combinatorics | Batch arrival | Nonexponential service times | Multi-server queue | STRONG APPROXIMATIONS | QUEUING-NETWORKS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | STABILIZING PERFORMANCE | MULTIPLE-CHANNEL QUEUES | TIME | Management research | Logarithms | Research | Batch processing | Queues (Computers) | Theorems | Operations research | Arrivals | Feedback | Stochastic processes | Variations | Queues | Queuing

Strong approximation | Customer feedback | Business and Management | Functional law of the iterated logarithm | Operations Research/Decision Theory | Theory of Computation | Combinatorics | Batch arrival | Nonexponential service times | Multi-server queue | STRONG APPROXIMATIONS | QUEUING-NETWORKS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | STABILIZING PERFORMANCE | MULTIPLE-CHANNEL QUEUES | TIME | Management research | Logarithms | Research | Batch processing | Queues (Computers) | Theorems | Operations research | Arrivals | Feedback | Stochastic processes | Variations | Queues | Queuing

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 6/2015, Volume 28, Issue 2, pp. 721 - 725

...J Theor Probab (2015) 28:721–725 DOI 10.1007/s10959-013-0481-4 The Limit Law of the Iterated Logarithm Xia Chen Received: 1 January 2013 / Revised: 8 February...

Brownian motion | The limit law of the iterated logarithm | 60G10 | 60G50 | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Ornstein–Uhlenbeck process | 60G15 | 60F15 | STATISTICS & PROBABILITY | Ornstein-Uhlenbeck process

Brownian motion | The limit law of the iterated logarithm | 60G10 | 60G50 | Probability Theory and Stochastic Processes | Mathematics | Statistics, general | Ornstein–Uhlenbeck process | 60G15 | 60F15 | STATISTICS & PROBABILITY | Ornstein-Uhlenbeck process

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 3/2018, Volume 31, Issue 1, pp. 579 - 597

...J Theor Probab (2018) 31:579–597 https://doi.org/10.1007/s10959-016-0710-8 An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics...

Law of the iterated logarithm | Secondary: 60G22 | Probability Theory and Stochastic Processes | 60G70 | Mathematics | Extremes of Gaussian processes | Order statistics process | Statistics, general | Primary: 60F15 | Gaussian processes

Law of the iterated logarithm | Secondary: 60G22 | Probability Theory and Stochastic Processes | 60G70 | Mathematics | Extremes of Gaussian processes | Order statistics process | Statistics, general | Primary: 60F15 | Gaussian processes

Journal Article

Annales de l'institut Henri Poincare (B) Probability and Statistics, ISSN 0246-0203, 08/2015, Volume 51, Issue 3, pp. 1124 - 1130

We prove Kolmogorov's law of the iterated logarithm for noncommutative martingales...

Quantum martingales | Noncommutative martingales | Exponential inequality | Law of the iterated logarithm | STATISTICS & PROBABILITY | 46L53 | 60F15

Quantum martingales | Noncommutative martingales | Exponential inequality | Law of the iterated logarithm | STATISTICS & PROBABILITY | 46L53 | 60F15

Journal Article

Asymptotic Analysis, ISSN 0921-7134, 03/2016, Volume 97, Issue 1-2, pp. 91 - 112

The law of the iterated logarithm for some Markov operators, which converge exponentially to the invariant measure, is established...

law of the iterated logarithm | Markov operators | MATHEMATICS, APPLIED | CONVERGENCE | CHAINS | DIVISION | Mathematics - Probability

law of the iterated logarithm | Markov operators | MATHEMATICS, APPLIED | CONVERGENCE | CHAINS | DIVISION | Mathematics - Probability

Journal Article

Journal of theoretical probability, ISSN 1572-9230, 2014, Volume 29, Issue 1, pp. 32 - 47

In this paper, we establish the analogue of Kolmogorov-type law of the logarithm for an array of independent random variables...

Kolmogorov’s inequality | Kolmogorov’s law of the logarithm | Probability Theory and Stochastic Processes | Mathematics | Bernstein’s inequality | Statistics, general | Independent random variables | 60F15 | Bernstein's inequality | LARGE NUMBERS | Kolmogorov's inequality | CONVERGENCE | STATISTICS & PROBABILITY | ITERATED LOGARITHM | Kolmogorov's law of the logarithm | RANDOM-VARIABLES | Information science

Kolmogorov’s inequality | Kolmogorov’s law of the logarithm | Probability Theory and Stochastic Processes | Mathematics | Bernstein’s inequality | Statistics, general | Independent random variables | 60F15 | Bernstein's inequality | LARGE NUMBERS | Kolmogorov's inequality | CONVERGENCE | STATISTICS & PROBABILITY | ITERATED LOGARITHM | Kolmogorov's law of the logarithm | RANDOM-VARIABLES | Information science

Journal Article

Journal of Theoretical Probability, ISSN 0894-9840, 12/2018, Volume 31, Issue 4, pp. 2366 - 2375

...) for the largest eigenvalue of a GUE matrix recently put forward by E. Paquette and O. Zeitouni. The proof relies on sharp tail bounds and superadditivity, close to the standard law of the iterated logarithm...

Tracy–Widom distribution | Law of the iterated logarithm | Directed last passage percolation | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Tail inequalities | 60F15 | STATISTICS & PROBABILITY | Tracy-Widom distribution

Tracy–Widom distribution | Law of the iterated logarithm | Directed last passage percolation | Probability Theory and Stochastic Processes | 60B20 | Mathematics | Statistics, general | Tail inequalities | 60F15 | STATISTICS & PROBABILITY | Tracy-Widom distribution

Journal Article

JOURNAL OF INEQUALITIES AND APPLICATIONS, ISSN 1029-242X, 04/2018, Volume 2018, Issue 1, pp. 1 - 18

.... In this paper, we give a local law of the iterated logarithm of the form lim sup(s down arrow 0) vertical bar S-t+S(H) - S-t(H)vertical bar/S-H root 2 log...

MATHEMATICS | MATHEMATICS, APPLIED | LIMIT-THEOREMS | Phi-variation | Iterated logarithm | SYSTEMS | TIME | Sub-fractional Brownian motion | PATH PROPERTIES | Brownian movements | Theorems | 60G22 | Φ-variation | Research | 60F17 | 60G15

MATHEMATICS | MATHEMATICS, APPLIED | LIMIT-THEOREMS | Phi-variation | Iterated logarithm | SYSTEMS | TIME | Sub-fractional Brownian motion | PATH PROPERTIES | Brownian movements | Theorems | 60G22 | Φ-variation | Research | 60F17 | 60G15

Journal Article

Journal of applied probability, ISSN 0021-9002, 12/2012, Volume 49, Issue 4, pp. 978 - 989

...–Uhlenbeck process with linear drift, such as the law of the iterated logarithm (LIL) and Berry–Esseen bounds. As an application of the Berry...

Maximum likelihood estimator | Precise rate | Ornstein-Uhlenbeck process | Berry-Esseen bound | Law of the iterated logarithm | maximum likelihood estimator | law of the iterated logarithm | 62N02 | 60G50 | Ornstein--Uhlenbeck process | Berry--Esseen bound | precise rate | 60F15

Maximum likelihood estimator | Precise rate | Ornstein-Uhlenbeck process | Berry-Esseen bound | Law of the iterated logarithm | maximum likelihood estimator | law of the iterated logarithm | 62N02 | 60G50 | Ornstein--Uhlenbeck process | Berry--Esseen bound | precise rate | 60F15

Journal Article

Probability Theory and Related Fields, ISSN 0178-8051, 6/2015, Volume 162, Issue 1, pp. 365 - 409

... satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space...

Tent maps | Mathematical and Computational Biology | Theoretical, Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Mathematics | Almost sure invariance principle | Transversality | Quantitative Finance | 37E05 | 37A10 | Statistics for Business/Economics/Mathematical Finance/Insurance | Operations Research/Decision Theory | Law of iterated logarithm | Interval maps | One-parameter families | 60F17 | NUMBERS | STATISTICS & PROBABILITY | UNIMODAL MAPS | EXPANDING MAPS | DYNAMICS | SYSTEMS | CONVERGENCE | PIECEWISE MONOTONIC TRANSFORMATIONS | ERGODIC-THEORY | POINT | Studies | Probability | Mathematical models

Tent maps | Mathematical and Computational Biology | Theoretical, Mathematical and Computational Physics | Probability Theory and Stochastic Processes | Mathematics | Almost sure invariance principle | Transversality | Quantitative Finance | 37E05 | 37A10 | Statistics for Business/Economics/Mathematical Finance/Insurance | Operations Research/Decision Theory | Law of iterated logarithm | Interval maps | One-parameter families | 60F17 | NUMBERS | STATISTICS & PROBABILITY | UNIMODAL MAPS | EXPANDING MAPS | DYNAMICS | SYSTEMS | CONVERGENCE | PIECEWISE MONOTONIC TRANSFORMATIONS | ERGODIC-THEORY | POINT | Studies | Probability | Mathematical models

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 11/2007, Volume 53, Issue 11, pp. 4383 - 4391

In the cryptanalysis of stream ciphers and pseudorandom sequences, the notions of linear, jump, and 2-adic complexity arise naturally to measure the...

LÉvy classes | linear complexity | 2 -adic complexity | pseudorandom sequences | Electronic mail | stream ciphers | Galois fields | law of the iterated logarithm | Approximation algorithms | AFSR | FCSR | Random sequences | isometry | jump complexity | Pseudorandom sequences | Isometry | Jump complexity | Stream ciphers | Law of the iterated logarithm | 2-adic complexity | Linear complexity | Lévy classes | FEEDBACK SHIFT REGISTERS | ALGORITHM | COMPUTER SCIENCE, INFORMATION SYSTEMS | TIME | ADIC NUMBERS | ENGINEERING, ELECTRICAL & ELECTRONIC | Levy classes | Sequences (Mathematics) | Usage | Algorithms | Logarithms | Research | Properties | Information theory | Approximation | Law | Mathematical analysis | Registers | Complexity

LÉvy classes | linear complexity | 2 -adic complexity | pseudorandom sequences | Electronic mail | stream ciphers | Galois fields | law of the iterated logarithm | Approximation algorithms | AFSR | FCSR | Random sequences | isometry | jump complexity | Pseudorandom sequences | Isometry | Jump complexity | Stream ciphers | Law of the iterated logarithm | 2-adic complexity | Linear complexity | Lévy classes | FEEDBACK SHIFT REGISTERS | ALGORITHM | COMPUTER SCIENCE, INFORMATION SYSTEMS | TIME | ADIC NUMBERS | ENGINEERING, ELECTRICAL & ELECTRONIC | Levy classes | Sequences (Mathematics) | Usage | Algorithms | Logarithms | Research | Properties | Information theory | Approximation | Law | Mathematical analysis | Registers | Complexity

Journal Article

SpringerPlus, ISSN 2193-1801, 12/2014, Volume 3, Issue 1, pp. 1 - 7

.... We say X satisfies the (α,β)-Chover-type law of the iterated logarithm (and write X∈C T L I L(α,β)) if almost surely. This paper is devoted to a characterization of X...

Sums of . random variables | ( α , β )-Chover-type law of the iterated logarithm | Symmetric stable distribution with exponent α | Science, general | (α,β)-Chover-type law of the iterated logarithm | Sums of i.i.d. random variables | WEIGHTED SUMS | LIMITING BEHAVIOR | (alpha, beta)-Chover-type law of the iterated logarithm | MULTIDISCIPLINARY SCIENCES | Symmetric stable distribution with exponent alpha | STABLE SUMMANDS

Sums of . random variables | ( α , β )-Chover-type law of the iterated logarithm | Symmetric stable distribution with exponent α | Science, general | (α,β)-Chover-type law of the iterated logarithm | Sums of i.i.d. random variables | WEIGHTED SUMS | LIMITING BEHAVIOR | (alpha, beta)-Chover-type law of the iterated logarithm | MULTIDISCIPLINARY SCIENCES | Symmetric stable distribution with exponent alpha | STABLE SUMMANDS

Journal Article

Queueing Systems, ISSN 0257-0130, 4/2015, Volume 79, Issue 3, pp. 251 - 291

A law of iterated logarithm (LIL) is established for a multiclass queueing model, having a preemptive priority service discipline, one server and $$K...

Non-Markovian queues | Strong approximation | 60J65 | Priority queues | Systems Theory, Control | Probability Theory and Stochastic Processes | Preemptive-resume discipline | Economics / Management Science | Multiclass queues | 90B22 | Operations Research/Decision Theory | Production/Logistics/Supply Chain | Law of iterated logarithm | 60K25 | Computer Communication Networks | 60F15 | STRONG APPROXIMATIONS | TIME | TRAFFIC LIMIT-THEOREM | QUEUING-NETWORKS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | Studies | Algorithms | Queuing | Priorities | Law | Approximation | Asymptotic properties | Texts | Queues | Preempting | Logarithms | Servers

Non-Markovian queues | Strong approximation | 60J65 | Priority queues | Systems Theory, Control | Probability Theory and Stochastic Processes | Preemptive-resume discipline | Economics / Management Science | Multiclass queues | 90B22 | Operations Research/Decision Theory | Production/Logistics/Supply Chain | Law of iterated logarithm | 60K25 | Computer Communication Networks | 60F15 | STRONG APPROXIMATIONS | TIME | TRAFFIC LIMIT-THEOREM | QUEUING-NETWORKS | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | Studies | Algorithms | Queuing | Priorities | Law | Approximation | Asymptotic properties | Texts | Queues | Preempting | Logarithms | Servers

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 06/2010, Volume 366, Issue 2, pp. 435 - 443

In this paper, a Chover-type law of the k-iterated logarithm is established for over(ρ, ̃)-mixing sequences of identically distributed random variables...

Law of the k-iterated logarithm | over(ρ, ̃)-Mixing sequence of random variables | Domain of attraction

Law of the k-iterated logarithm | over(ρ, ̃)-Mixing sequence of random variables | Domain of attraction

Journal Article

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