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Concept in statistics
physics, statistics, econometrics and signal processing, a stochastic process is said to be in an ergodic regime if an observable's ensemble average equals
Ergodic_process
Property of measure-preserving dynamical systems
In mathematics, especially in ergodic theory, ergodicity is a way of saying that a dynamical system behaves as one indivisible statistical system, rather
Ergodicity
Stochastic process that exhibits both stationarity and ergodicity
stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the random process will not
Stationary_ergodic_process
Branch of mathematics that studies dynamical systems
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this
Ergodic_theory
Statistical mechanics hypothesis that all microstates are equiprobable for a given energy
group-level data. Ergodic process Ergodic theory, a branch of mathematics concerned with a more general formulation of ergodicity Ergodicity Loschmidt's paradox
Ergodic_hypothesis
Literary genre
Ergodic literature is a mode of textual organization in which nontrivial effort is required for the reader to traverse the text, beyond ordinary eye movement
Ergodic_literature
Topic in mathematics
|\Omega |<\infty } ) stationary ergodic stochastic processes in the Shannon–McMillan–Breiman theorem using the ergodic theory and for any i.i.d. sources
Asymptotic equipartition property
Asymptotic_equipartition_property
Class of stochastic process
{\displaystyle X_{t}} does not converge since the process is not ergodic. As a further example of a stationary process for which any single realisation has an apparently
Stationary_process
Topics referred to by the same term
Aside from its generic use as the generic adjective ergodic, ergodic may relate to: Ergodicity, mathematical description of a dynamical system which, broadly
Ergodic_(disambiguation)
Random process independent of past history
fact, merely irreducible Markov chains correspond to ergodic processes, defined according to ergodic theory. Some authors call a matrix primitive if there
Markov_chain
Collection of random variables
stochastic processes topics Covariance function Deterministic system Dynamics of Markovian particles Entropy rate (for a stochastic process) Ergodic process Gillespie
Stochastic_process
Integer sequence in mathematics
^{\omega }} is ergodic. Fibonacci numbers are not an ergodic sequence. Ergodic theory Ergodic process, for the use of the term in signal processing See, generally
Ergodic_sequence
Theory that attempts to blend economics and ergodic theory
interested in understanding how behaviour is shaped by non-ergodic economic processes, that is processes where the expectation value of an observable does not
Ergodicity_economics
Representation of a type of random process
a modelled representation of a type of random process. It can be used to describe time-varying processes from many natural and artificial sources. The
Autoregressive_model
Russian chemist
"Electron Capture Dissociation of Multiply Charged Protein Cations. A Non-ergodic Process". Journal of the American Chemical Society. 120 (13): 3265–3266. Bibcode:1998JAChS
Roman_Zubarev
NP-hard problem in combinatorial optimization
X_{1},\ldots ,X_{n}} are replaced with observations from a stationary ergodic process with uniform marginals. One has L ∗ ≤ 2 n + 2 {\displaystyle L^{*}\leq
Travelling_salesman_problem
Stochastic process modeling random walk with friction
(2021-03-19). "Noise and ergodic properties of Brownian motion in an optical tweezer: Looking at regime crossovers in an Ornstein-Uhlenbeck process". Physical Review
Ornstein–Uhlenbeck_process
Index of articles associated with the same name
related to eigenvectors for which the eigenvalue is unity. Stationary ergodic process Perron–Frobenius theorem Stationary state or ground state in quantum
Stationary_distribution
subsets taken simultaneously. Difficulty is commonly avoided using an ergodic process, that changes with time and probability gets involved with it, and
Blackman–Tukey_transformation
Correlation of a signal with a time-shifted copy of itself, as a function of shift
processes that are not stationary, these will also be functions of t {\displaystyle t} , or n {\displaystyle n} . For processes that are also ergodic
Autocorrelation
Power law growth of entropy of language or a stochastic process
information measure. A perigraphic process is roughly an algorithmically random ergodic component of a non-ergodic process with a non-atomic invariant sigma-algebra
Hilberg's_hypothesis
Mathematical description of mixing substances
thermodynamic process of mixing in the everyday world: e.g. mixing paint, mixing drinks, mixing metals. The concept appears in ergodic theory—the study
Mixing_(mathematics)
In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been
Ergodic_flow
Type of signal processing statistic
transfer between input and output of a linear system. If the signals are ergodic, and the system function is linear, it can be used to estimate the causality
Coherence_(signal_processing)
Solution to a stochastic differential equation
statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion processes are stochastic
Diffusion_process
Mathematical model for sequential decision making under uncertainty
approaches applied. Here we only consider the ergodic model, which means our continuous-time MDP becomes an ergodic continuous-time Markov chain under a stationary
Markov_decision_process
Special type of continuous-time Markov process
_{n=1}^{i}{\frac {\mu _{n}}{\lambda _{n}}}=\infty .} A birth-and-death process is ergodic if and only if ∑ i = 1 ∞ ∏ n = 1 i μ n λ n = ∞ and ∑ i = 1 ∞ ∏ n =
Birth–death_process
Sequence of data points over time
conditions under which much of the theory is built: Stationary process Ergodic process Ergodicity implies stationarity, but the converse is not necessarily
Time_series
Random process of binary (boolean) random variables
{\displaystyle \mathbb {Z} ^{x}} are ergodic sequences.[verification needed] From any Bernoulli process one may derive a Bernoulli process with p = 1/2 by the von
Bernoulli_process
Theory of probability
Francesco Cantelli, in 1933. If X n {\displaystyle X_{n}} is a stationary ergodic process, then F n ( x ) {\displaystyle F_{n}(x)} converges almost surely to
Glivenko–Cantelli_theorem
Path selection method for aircraft
and navigation system error (NSE). It is assumed that FTE is an ergodic stochastic process within a given flight control mode. As a result, the FTE distribution
Required navigation performance
Required_navigation_performance
Romanian-American mathematician (1935–2025)
Romanian-American mathematician who made contributions to the fields of ergodic theory, probability and analysis. Bellow was born in Bucharest, Romania
Alexandra_Bellow
Formula in probability theory
theory, Rice's formula counts the average number of times an ergodic stationary process X(t) per unit time crosses a fixed level u. Adler and Taylor describe
Rice's_formula
Type of set in information theory
source. The AEP can also be proven for a large class of stationary ergodic processes, allowing typical set to be defined in more general cases. Additionally
Typical_set
Process that cannot be undone or reversed
In thermodynamics, an irreversible process is a process impossible to reverse or undo. All complex natural processes are irreversible, although a phase
Irreversible_process
the process is repeated, the waves become more and more focused on the target. Yet another variation is to use a single transducer and an ergodic cavity
Time reversal signal processing
Time_reversal_signal_processing
Kingman's subadditive ergodic theorem is one of several ergodic theorems. It can be seen as a generalization of Birkhoff's ergodic theorem. Intuitively
Kingman's subadditive ergodic theorem
Kingman's_subadditive_ergodic_theorem
Type of shift space studied in ergodic theory
systems, and in particular are objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by
Subshift_of_finite_type
statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process for which the index variable takes a
Continuous-time stochastic process
Continuous-time_stochastic_process
Stochastic volatility model used in derivatives markets
{\displaystyle \max(F_{T}-K,\;0)} under the probability distribution of the process F t {\displaystyle F_{t}} . Except for the special cases of β = 0 {\displaystyle
SABR_volatility_model
Mathematical model of the time dependence of a point in space
Lai-Sang Young Ergodic Theory A Probabilistic Approach to Dynamical Systems, Ch3, ergodic theorems Alex Blumenthal Lai-Sang Young Ergodic Theory A Probabilistic
Dynamical_system
(image processing) Epps effect Equating – test equating Equipossible Equiprobable Erdős–Rényi model Erlang distribution Ergodic theory Ergodicity Error
List_of_statistics_articles
Computing using random bit streams
stochastic processing. Ergodic Processing involves sending a stream of bundles, which captures the benefits of regular stochastic and bundle processing. Burst
Stochastic_computing
Concept in quantum optics
pulses, the ensemble is made up of many pulses. When one deals with ergodic processes, the ensemble average can be replaced with a time average. If we restrict
Higher_order_coherence
2000 novel by Mark Z. Danielewski
points, the book must be rotated to be read, making it a prime example of ergodic literature. The book is most often described as a horror story, though
House_of_Leaves
Quantum analog of probabilistic automata
arbitrarily sharp transformations, but rather as an ergodic process, or more accurately, a mixing process that only concatenates transformations onto a state
Quantum_finite_automaton
Concept in statistics
functions of the variables. A one-dimensional GRF is also called a Gaussian process. An important special case of a GRF is the Gaussian free field. With regard
Gaussian_random_field
Costa, O. L. V.; Dufour, F. (2008). "Stability and Ergodicity of Piecewise Deterministic Markov Processes" (PDF). SIAM Journal on Control and Optimization
Piecewise-deterministic Markov process
Piecewise-deterministic_Markov_process
Mathematical lemma in ergodic theory
In ergodic theory, Kac's lemma, demonstrated by mathematician Mark Kac in 1947, is a lemma stating that in a finite measure space the orbit of almost
Kac's_lemma
Probability concept
One method of finding the stationary probability distribution, π, of an ergodic continuous-time Markov chain, Q, is by first finding its embedded Markov
Continuous-time_Markov_chain
American mathematician
in probability theory on graphs, combinatorics, statistical mechanics, ergodic theory and harmonic analysis. Lyons graduated with B.A. mathematics in
Russell_Lyons
process Coupling (probability) Ergodic theory Maximal ergodic theorem Ergodic (adjective) Galton–Watson process Gauss–Markov process Gaussian process
List_of_probability_topics
Type of motion in mechanical engineering
approaches. Mathematically, random vibration is characterized as an ergodic and stationary process. The acceleration spectral density (ASD) or power spectral density
Random_vibration
Stochastic model for the evolution of financial interest rates
^{2}\,} , the Feller square-root process can be obtained from the square of an Ornstein–Uhlenbeck process. It is ergodic and possesses a stationary distribution
Cox–Ingersoll–Ross_model
stationary stochastic processes, including Markov chains and subshifts of finite type, Anosov flows and Sinai's billiards, ergodic automorphisms of the
Ornstein_isomorphism_theorem
Subject of study in ergodic theory
object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence
Measure-preserving dynamical system
Measure-preserving_dynamical_system
Type of interactive fiction
Cybertext as defined by Espen Aarseth in 1997 is a type of ergodic literature where the user traverses the text by doing nontrivial work. Cybertexts are
Cybertext
American-Israeli mathematician
areas such as number theory, fractals, signal processing and electrical engineering. In 1977, he gave an ergodic theory reformulation, and subsequently proof
Hillel_Furstenberg
Narrative with imaginary elements
Fiction writing is the process by which an author or creator produces a fictional work. Some elements of the writing process may be planned in advance
Fiction
Type of vector space in math
Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics)
Hilbert_space
Discipline that studies financial markets as physical systems
addresses issues such as theory of price formation, price dynamics, market ergodicity, collective phenomena, market self-action, and market instabilities. Physics
Physics_of_financial_markets
Genre of electronic literature
is a kind of ergodic literature: In ergodic literature, nontrivial effort is required to allow the reader to traverse the text. If ergodic literature is
Hypertext_fiction
Process of mechanically stirring a heterogeneous mixture to homogenize it
effect. The mathematics of mixing is highly abstract, and is a part of ergodic theory, itself a part of chaos theory. The type of operation and equipment
Mixing_(process_engineering)
American mathematician (1884–1944)
problem, and general relativity. Today, Birkhoff is best remembered for the ergodic theorem. The George D. Birkhoff House, his residence in Cambridge, Massachusetts
George_David_Birkhoff
Hungarian and American mathematician and physicist (1903–1957)
to ergodic theory, a branch of mathematics that involves the states of dynamical systems with an invariant measure. Of the 1932 papers on ergodic theory
John_von_Neumann
Russian–American mathematician (born 1935)
Random Processes (with Koralov). 2nd edition, Springer, 2007. Theory of Phase Transitions – Rigorous Results. Pergamon, Oxford 1982. Ergodic Theory (with
Yakov_Sinai
Theorem in queueing theory
as well as the whole thing. The only requirement is that the system be ergodic. In some cases it is possible not only to mathematically relate the average
Little's_law
Spectral density estimation technique
It can estimate the power spectrum SX of a stationary ergodic finite-variance random process X, given a finite contiguous realization of X as data. The
Multitaper
Probability concept
S}q_{ij}k_{j}^{A}=1&{\text{ for }}i\notin A.\end{aligned}}} An instance of ergodic theory, the ergodic theorem states that for an irreducible aperiodic Markov chain
Discrete-time_Markov_chain
fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation
Markov_odometer
Algorithm in statistical physics
should produce the same distribution, as long as the algorithm satisfies ergodicity and detailed balance. In both algorithms, for any change in energy, p
Glauber_dynamics
Argentine-American mathematician
from interpolation theory to Cauchy integrals on Lipschitz curves, from ergodic theory to inverse problems in electrical prospection. Calderón's work has
Alberto_Calderón
Type of stochastic process
with the continuum notions of shape. The jump-diffusion process was constructed to have ergodic properties so that after initially flowing away from its
Jump_diffusion
American author (born 1966)
page and the reader. Early on, critics characterized his writing as being ergodic literature, and Danielewski has described his style as: Signiconic = sign
Mark_Z._Danielewski
term that covers the law of large numbers, all central limit theorems and ergodic theorems. If one throws a dice once, it is difficult to predict the outcome
Statistical_regularity
Physical law for entropy and heat
statement was shown to be equivalent to the statement of Clausius. The ergodic hypothesis is also important for Ludwig Boltzmann's approach. It says that
Second_law_of_thermodynamics
the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. It states that an aperiodic measure preserving dynamical system
Rokhlin_lemma
Disordered magnetic state
describing the slow dynamics of the magnetization and the complex non-ergodic equilibrium state. Unlike the Edwards–Anderson (EA) model, the range of
Spin_glass
Use of the second law of thermodynamics to distinguish past from future
many of the interesting cases are either ergodic or mixing, and it is strongly suspected that mixing and ergodicity somehow underlie the fundamental mechanism
Entropy_as_an_arrow_of_time
Relative importance of certain frequencies in a composite signal
{\displaystyle R_{xx}(\tau )} , provided that x ( t ) {\displaystyle x(t)} is ergodic, which is true in most, but not all, practical cases. lim T → ∞ 1 T | x
Spectral_density
in equilibrium can be modeled, via the ergodic hypothesis, as the stationary distribution of a stochastic process. Then the Euclidean path integral measure
Stochastic_quantization
Generalization of the Bernoulli process to more than two possible outcomes
Press (1973) Michael S. Keane, "Ergodic theory and subshifts of finite type", (1991), appearing as Chapter 2 in Ergodic Theory, Symbolic Dynamics and Hyperbolic
Bernoulli_scheme
Mathematical Theory
is known to ensure similar performance guarantees for more general ergodic processes ω ( t ) {\displaystyle \omega (t)} , so that the i.i.d. assumption
Drift_plus_penalty
Israeli mathematician
articles in the field of probability concentrating on Markov chains and ergodic theory. He serves as professor emeritus at the Department of Mathematics
Michael_Lin_(mathematician)
Diffusion process with a non-linear relationship to time
Simon, Blair; Tamkun, Michael M.; Krapf, Diego (2011-04-19). "Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule
Anomalous_diffusion
Use of multiple antennas in radio
and the noise vector, respectively. Referring to information theory, the ergodic channel capacity of MIMO systems where both the transmitter and the receiver
MIMO
Lossless data compression algorithms
entropic—If X {\textstyle X} is a binary source that is stationary and ergodic, then lim sup n 1 n l L Z 78 ( X 1 : n ) ≤ h ( X ) {\displaystyle \limsup
LZ77_and_LZ78
Random motion of particles suspended in a fluid
Cohen, Ruben D. (1986). "Self Similarity in Brownian Motion and Other Ergodic Phenomena" (PDF). Journal of Chemical Education. 63 (11): 933–934. Bibcode:1986JChEd
Brownian_motion
American mathematician
mathematician best known for his work in information theory, ergodic theory, and stationary process theory. Kieffer received his elementary and high school
John_Kieffer
Left-invariant (or right-invariant) measure on locally compact topological group
group theory, representation theory, statistics, probability theory, and ergodic theory. Let ( G , ⋅ ) {\displaystyle (G,\cdot )} be a locally compact Hausdorff
Haar_measure
Physics of many interacting particles
arguments in favour of the equal a priori probability postulate: Ergodic hypothesis: An ergodic system is one that evolves over time to explore "all accessible"
Statistical_mechanics
Theorem in Ramsey theory
Furstenberg and Weiss proved an equivalent form of the theorem in 1978, using ergodic theory. multiple Birkhoff recurrence theorem (Furstenberg and Weiss, 1978)—If
Van_der_Waerden's_theorem
Philosophical thought experiment
an otherwise featureless universe. In the universe's eventual state of ergodic "heat death", given enough time, every possible structure (including every
Boltzmann_brain
One of two theorems in dynamical systems
For the 1st theorem: Ya. G. Sinai (Ed.) (1997): Dynamical Systems II. Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics
Krylov–Bogolyubov_theorem
Sender of a communication
possible elsewhere in space or time. A source may be modelled as memoryless, ergodic, stationary, or stochastic, in order of increasing generality.[citation
Communication_source
Theorem in linear algebra
matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite
Perron–Frobenius_theorem
Area of mathematics
in part it deals with influencing the behavior of dynamical systems. Ergodic theory is a branch of mathematics that studies dynamical systems with an
Dynamical_systems_theory
Base of natural logarithms
distinguished role in the theory of entropy in probability theory and ergodic theory. The basic idea is to consider a partition of a probability space
E_(mathematical_constant)
Probabilistic problem-solving algorithm
method will be samples from the desired (target) distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures
Monte_Carlo_method
Bet sizing formula for long-term growth
real life). The debate was renewed by evoking ergodicity breaking. Yet the difference between ergodicity breaking and Knightian uncertainty should be recognized
Kelly_criterion
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