AI & ChatGPT searches , social queries for ALMOST SURELY
Search references for ALMOST SURELY. Phrases containing ALMOST SURELY
See searches and references containing ALMOST SURELY!
Search references for ALMOST SURELY. Phrases containing ALMOST SURELY
See searches and references containing ALMOST SURELY!ALMOST SURELY
Probability saying
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the
Almost_surely
Probability distribution
.} A constant random variable is almost surely constant, but the converse is not true, since if X is almost surely constant then there may still exist
Degenerate_distribution
Sigma-algebra used in probability and ergodic theory
{\mathcal {F}}} is called almost surely invariant if and only if its indicator function 1 S {\displaystyle 1_{S}} is almost surely equal to the indicator
Invariant_sigma-algebra
Counterintuitive result in probability
random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, including the complete works of William Shakespeare
Infinite_monkey_theorem
Notions of probabilistic convergence, applied to estimation and asymptotic analysis
elementary real analysis. To say that the sequence Xn converges almost surely or almost everywhere or with probability 1 or strongly towards X {\displaystyle
Convergence of random variables
Convergence_of_random_variables
Everywhere except a set of measure zero
complement is of measure zero. In probability theory, the terms almost surely, almost certain and almost always refer to events with probability 1 not necessarily
Almost_everywhere
Averages of repeated trials converge to the expected value
flips will almost surely converge to 1⁄2 as n approaches infinity. Although the proportion of heads (and tails) approaches 1⁄2, almost surely the absolute
Law_of_large_numbers
Stochastic process generalizing Brownian motion
normally distributed with mean 0 and variance u. W has almost surely continuous paths: Wt is almost surely continuous in t. That the process has independent
Wiener_process
In mathematics, with negligible exceptions
generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in probability
Almost_all
Theorem in probability theory
\}}|\leq c} almost surely for all t ∈ N 0 {\displaystyle t\in \mathbb {N} _{0}} . Then X τ {\displaystyle X_{\tau }} is an almost surely well-defined
Optional_stopping_theorem
Term in set theory
Almost periodic function - and Operators Almost all Almost surely Approximation List of mathematical jargon "Almost All Real Numbers are Transcendental -
Almost
Process forming a path from many random steps
points have infinitely many intersections almost surely, but for dimensions higher than 5, they almost surely intersect only finitely often. The asymptotic
Random_walk
Tree-based ensemble machine learning methods
{\displaystyle \mathbf {x} } and z {\displaystyle \mathbf {z} } , then almost surely we have m ~ M , n ( x , Θ 1 , … , Θ M ) = ∑ i = 1 n Y i K M , n ( x
Random_forest
Probabilistic inequality applying on sum of bounded random variables
such that a i ≤ X i ≤ b i {\displaystyle a_{i}\leq X_{i}\leq b_{i}} almost surely. Consider the sum of these random variables, S n = X 1 + ⋯ + X n . {\displaystyle
Hoeffding's_inequality
Self-adjoint operator that arises in physical transition problems
\alpha }} has almost surely pure point spectrum and exhibits Anderson localization. (It is known that almost surely can not be replaced by surely.) That the
Almost_Mathieu_operator
dimension of a set A {\displaystyle A} under a Brownian motion doubles almost surely. The first result is due to Henry P. McKean jr and hence called McKean's
Dimension_doubling_theorem
Model in probability theory
X_{n}\longrightarrow X_{\infty }} almost surely. In particular, every nonnegative supermartingale converges almost surely, and its limiting expectation satisfies
Martingale (probability theory)
Martingale_(probability_theory)
Differential equations involving stochastic processes
discovered to be exceptionally complex mathematically. The Wiener process is almost surely nowhere differentiable; thus, it requires its own rules of calculus
Stochastic differential equation
Stochastic_differential_equation
Lemma in measure theory
]}\leq \liminf _{n\to \infty }\,\mathbb {E} [X_{n}|{\mathcal {G}}]} almost surely. Note: Conditional expectation for non-negative random variables is
Fatou's_lemma
Two closely related models for generating random graphs
G(n, p) will almost surely have no connected components of size larger than O(log(n)). If np = 1, then a graph in G(n, p) will almost surely have a largest
Erdős–Rényi_model
Branch of mathematics that studies dynamical systems
) (almost surely) {\displaystyle {\frac {R_{1}+\cdots +R_{n}}{n}}\to {\frac {\mu (X)}{\mu (A)}}\quad {\text{(almost surely)}}} (See almost surely.) That
Ergodic_theory
Class of flowering plants
name for a clade). Therefore, in practice the name Liliopsida will almost surely refer to the usage as in the Cronquist system. In summary the monocotyledons
Liliopsida
When the occurrence of one event does not affect the likelihood of another
event is independent of itself if and only if it almost surely occurs or its complement almost surely occurs; this fact is useful when proving zero–one
Independence (probability theory)
Independence_(probability_theory)
Theorems concerning stochastic processes
{\textstyle X_{t}^{-}=-\min(X_{t},0)} . Then the sequence converges almost surely to a random variable X {\displaystyle X} with finite expectation. There
Doob's martingale convergence theorems
Doob's_martingale_convergence_theorems
Variety of proofs provided for the different types of convergence of random variables
{ X n } {\displaystyle \{X_{n}\}} converges to X {\displaystyle X} almost surely, it means that the set of points O = { ω : lim X n ( ω ) ≠ X ( ω ) }
Proofs of convergence of random variables
Proofs_of_convergence_of_random_variables
Mathematical theorem
logarithm, "lim sup" denotes the limit superior, and "a.s." stands for "almost surely". Another statement given by A. N. Kolmogorov in 1929 is as follows
Law_of_the_iterated_logarithm
Decomposition of a graph into hamiltonion cycles
easy Hamiltonian cycle problems. Random regular graphs of even degree almost surely have a Hamiltonian decomposition, and more strongly there is a randomized
Hamiltonian_decomposition
Variable representing a random phenomenon
considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution. In increasing order of strength, the precise
Random_variable
Average value of a random variable
"almost surely"—a central property of the Lebesgue integral. Basically, one says that an inequality like X ≥ 0 {\displaystyle X\geq 0} is true almost surely
Expected_value
Random motion of particles suspended in a fluid
The Wiener process Wt is characterized by four facts: W0 = 0 Wt is almost surely continuous Wt has independent increments W t − W s ∼ N ( 0 , t − s )
Brownian_motion
Mathematical theorem in stochastic processes
processes, this implies iteratively that Yn = 0 almost surely for all n ∈ I, hence the decomposition is almost surely unique. A real-valued stochastic process
Doob_decomposition_theorem
Concept in probability theory
series ∑ n = 1 ∞ X n {\textstyle \sum _{n=1}^{\infty }X_{n}} converges almost surely in R {\displaystyle \mathbb {R} } if the following conditions hold for
Kolmogorov's three-series theorem
Kolmogorov's_three-series_theorem
Process for achieving system stability
critical probability below which all cells almost surely become inactive and above which some clusters almost surely survive. Bootstrap percolation can be
Bootstrap_percolation
sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions. Let (Ω, Σ, P) be a probability space. Let X : I × Ω → S
Sample-continuous_process
Branch of probability theory
{1}{t}}Y_{t}={\frac {1}{\operatorname {E} [S_{1}]}}\operatorname {E} [W_{1}]} almost surely. Renewal processes additionally have a property analogous to the central
Renewal_theory
Theorem in probability theory
that specifies that a certain type of event will either almost surely happen or almost surely not happen. It is sometimes known as the Savage-Hewitt law
Hewitt–Savage_zero–one_law
Type of stochastic process
Y t {\displaystyle Y_{t}} is continuous almost surely (since W 1 ≠ 0 {\displaystyle W_{1}\neq 0} almost surely), nevertheless, its expectation is discontinuous
Local_martingale
Theorem in probability theory
Hence, the probability of Xn = 0 occurring for infinitely many n is 0. Almost surely (i.e., with probability 1), Xn is nonzero for all but finitely many n
Borel–Cantelli_lemma
Function type in graph theory
the random graph models defined by graphons give rise to dense graphs almost surely, and, by the regularity lemma, graphons capture the structure of arbitrary
Graphon
measurability theorem. A function f {\displaystyle f} is said to be almost surely separably valued (or essentially separably valued) if there exists a
Weakly_measurable_function
Difficulties arising when analyzing data with many aspects ("dimensions")
blessing, as it ensures that the nearest-neighbor of an instance is almost-surely its most closely related instance. From this perspective, contrast-loss
Curse_of_dimensionality
Theory of probability
distribution function converges uniformly to the true distribution function almost surely. The uniform convergence of more general empirical measures becomes
Glivenko–Cantelli_theorem
Exponentially decreasing bounds on tail distributions of random variables
If ‖ M i ‖ ≤ γ {\displaystyle \lVert M_{i}\rVert \leq \gamma } holds almost surely for all i ∈ { 1 , … , t } {\displaystyle i\in \{1,\ldots ,t\}} , then
Chernoff_bound
Topic in mathematics
B,p)} , the asymptotic equipartition property is an assertion that, almost surely, − 1 n log p ( X 1 , X 2 , … , X n ) → H ( X ) as n → ∞ {\displaystyle
Asymptotic equipartition property
Asymptotic_equipartition_property
a property of a logic saying that any property is either almost surely true or almost surely false. Zero-one law holds for first-order logic (without
Zero–one_law_(logic)
Inequality in probability theory
real-valued random variable such that a ≤ X ≤ b {\displaystyle a\leq X\leq b} almost surely, i.e. with probability one. Then, for all λ ∈ R {\displaystyle \lambda
Hoeffding's_lemma
Probability theorem
the intersection of two almost sure events is almost sure. By definition, we conclude that g(Xn) converges to g(X) almost surely. Slutsky's theorem Portmanteau
Continuous_mapping_theorem
Variance of a random variable given value of other variables
) = Var ( Y | X ) {\displaystyle v(X)=\operatorname {Var} (Y|X)} almost surely. The "conditional expectation of Y given X=x" can also be defined more
Conditional_variance
Stochastic process that is a continuous function of time or index parameter
Xt. X is said to be sample continuous if Xt(ω) is continuous in t for P-almost all ω ∈ Ω. Sample continuity is the appropriate notion of continuity for
Continuous_stochastic_process
Generalization of volume to non-integer number of dimensions
infinite d {\displaystyle d} -dimensional Hausdorff measure. For example, almost surely the image of planar Brownian motion has Hausdorff dimension 2 and its
Hausdorff_measure
Topics referred to by the same term
reference work Aggregate supply, in economics, the total supply of goods Almost surely or a.s., in probability theory Aš, a town in the Czech Republic Ås,
As
Probability concept
trajectory ( X t ( ω ) ) t ∈ T {\displaystyle (X_{t}(\omega ))_{t\in T}} is almost surely right continuous (with respect to the discrete metric on S {\displaystyle
Continuous-time_Markov_chain
Probability of stochastic processes
|X_{t+1}-X_{t}|\leq k} almost surely for some fixed bound k and all t. Also assume that | X 1 | ≤ k {\displaystyle |X_{1}|\leq k} almost surely. Define σ t 2 =
Martingale central limit theorem
Martingale_central_limit_theorem
Range of views in political science and philosophy
The thesis predicted by Condorcet's jury theorem either occurs almost surely or almost never, and the prior probability of the theorem's thesis is zero
Epistemic_democracy
Theorem in probability theory
is not symmetric, then (9) does not hold. Note that, when Z is not almost surely equal to the zero random variable, then (11) and (12) cannot hold simultaneously
Wald's_equation
Technique where the user tests the application or system by providing random inputs
random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare
Monkey_testing
Statistical function that defines the quantiles of a probability distribution
function Q behaves as an "almost sure left inverse" for the distribution function, in the sense that Q ( F ( X ) ) = X almost surely. {\displaystyle Q{\bigl
Quantile_function
asymptotically almost surely. In particular, for r ≥ 3 {\displaystyle r\geq 3} , a random r-regular graph of large size is asymptotically almost surely r-connected
Random_regular_graph
Mathematical measure space associated to a random walk
identification can be made directly since a path in Wiener space converges almost surely to a point on ∂ D {\displaystyle \partial \mathbb {D} } ). This interpretation
Poisson_boundary
Stochastic process in probability theory
satisfies the following properties: X 0 = 0 {\displaystyle X_{0}=0\,} almost surely; Independence of increments: For any 0 ≤ t 1 < t 2 < ⋯ < t n < ∞ {\displaystyle
Lévy_process
Cryptographic model of a random function
two classes shown in 1981 to be distinct relative to a random oracle almost surely. They made their way into cryptography by the publication of Mihir Bellare
Random_oracle
Plan designed to achieve profitable return
strategy or admissible strategy is any trading strategy with wealth almost surely bounded from below. In particular, an admissible trading strategy precludes
Admissible_trading_strategy
subordinator a process must be a Lévy process. It also must be increasing, almost surely, or an additive process. A subordinator is a real-valued stochastic
Subordinator_(mathematics)
Type of random process
process that admits one or several absorbing states that are reached almost surely, but is initially distributed such that it can evolve for a long time
Quasi-stationary_distribution
K\colon \Omega \to 2^{M}} such that K ( ω ) {\displaystyle K(\omega )} is almost surely compact and ω ↦ inf b ∈ K ( ω ) d ( x , b ) {\displaystyle \omega \mapsto
Random_compact_set
for a random dynamical system is a P {\displaystyle \mathbb {P} } -almost surely unique random set such that A ( ω ) {\displaystyle {\mathcal {A}}(\omega
Pullback_attractor
hitting keys at random on typewriters for an infinite amount of time will almost surely produce Hamlet." The play's humour mainly involves literary references
Infinite monkey theorem in popular culture
Infinite_monkey_theorem_in_popular_culture
Category of painful thinking or beliefs
anytime... My sharp knife could kill a baby. This horrific thought is almost surely revealing. I am probably going to kill my child in the near future.
Thought-action_fusion
whereas for the almost-always complexity estimate, it is evaluated that the algorithm admits a given complexity estimate that almost surely holds. In probabilistic
Probabilistic analysis of algorithms
Probabilistic_analysis_of_algorithms
Random set of points on a space with random number and random position
elements of S. If X i {\displaystyle X_{i}} 's are almost surely distinct (or equivalently, almost surely ξ ( x ) ≤ 1 {\displaystyle \xi (x)\leq 1} for all
Point_process
Invariant measure of fractal dimension
space of dimension 2 and above have Hausdorff dimension equal to 2 almost surely. Lewis Fry Richardson performed detailed experiments to measure the
Hausdorff_dimension
Quantity defined for a stochastic process
the sense of the definition given here and its paths be nonetheless almost surely of infinite 1-variation for every t > 0 {\displaystyle t>0} in the classical
Quadratic_variation
sup t ∈ T | f t | {\displaystyle \|f\|_{T}:=\sup _{t\in T}|f_{t}|} almost surely finite, and let σ T 2 := sup t ∈ T E | f t | 2 . {\displaystyle \sigma
Borell–TIS_inequality
Mathematical theory
that a random variable of Jessen–Wintner type, meaning the sum of an almost surely convergent series of independent discrete random variables, is of pure
Jessen–Wintner_theorem
Statistical measure
of the linear subspaces spanned by X and Y samples respectively are almost surely equal and if we assume that these subspaces are equal, then in this
Distance_correlation
Probability equation in mathematics
{E} [Z]^{2}}}.} This inequality is sharp; equality is achieved if Z almost surely equals a positive constant. In turn, this implies another convenient
Paley–Zygmund_inequality
Time fuze used by British Special forces
batches be used together. That way if one detonator fails the other will almost surely blow the charge. Note that if both detonators were going to work, the
Pencil_detonator
Branch of mathematics concerning probability
all i, so that Y ¯ n {\displaystyle {\bar {Y}}_{n}} converges to p almost surely. The central limit theorem (CLT) explains the ubiquitous occurrence
Probability_theory
Randomized mathematical sequence based upon the Fibonacci sequence
decays almost surely if β is less than a critical value β* ≈ 0.70258, known as the Embree–Trefethen constant, and otherwise grows almost surely. They also
Random_Fibonacci_sequence
Unsolved problem in computer science
turn out to be helpful even if it is proved, because such a proof will almost surely be nonconstructive. A proof of P ≠ NP would lack the practical computational
P_versus_NP_problem
Method of estimating the parameters of a statistical model, given observations
{\widehat {\theta \,}}} converges to θ0 almost surely, then a stronger condition of uniform convergence almost surely has to be imposed: sup θ ∈ Θ ‖ ℓ ^ (
Maximum_likelihood_estimation
Mathematical statistics distance measure
which converges if and only if P ≤ 2 Q {\displaystyle P\leq 2Q} almost surely w.r.t Q {\displaystyle Q} . [Proof] Denote f ( α ) := D KL ( ( 1 − α
Kullback–Leibler_divergence
Collection of Srinivasa Ramanujan's discoveries in mathematics
where he probably received the manuscripts of the lost notebook. ... Almost surely, this manuscript, or at least most of it, was written during the last
Ramanujan's_lost_notebook
Political confrontation in Fascist Italy
Felice argued that Mussolini was a political victim of a plot, and almost surely he was damaged by the crisis that followed the murder. Many fascists
Matteotti_Crisis
Solution to a stochastic differential equation
diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion processes are stochastic in nature
Diffusion_process
References A – adele ring or algebraic numbers. a.a.s. – asymptotically almost surely. AC – Axiom of Choice, or set of absolutely continuous functions. a
List of mathematical abbreviations
List_of_mathematical_abbreviations
Number measuring the chance an event occurs
for instance, exactly 7 – is 0. This means that an observation will almost surely not be exactly 7. However, it does not mean that exactly 7 is impossible
Probability
Measure of the joint variability
\operatorname {cov} (X,X)=0} implies that X {\displaystyle X} is constant almost surely. In fact these properties imply that the covariance defines an inner
Covariance
Sorting algorithm
the right permutation, so given an unbounded number of tries it will almost surely eventually be chosen. Gorosort An algorithm introduced in the 2011 Google
Bogosort
Statistical model
_{n}c_{n}(|\xi _{n}|+|\eta _{n}|)<\infty } almost surely, which ensures uniform convergence of the Fourier series almost surely, and sample continuity of X . {\displaystyle
Gaussian_process
1851 novel by Herman Melville
boundaries by the solid realism of the whaling context. Ripley was almost surely also the author of the review in Harper's for December, which saw in
Moby-Dick
Optimization algorithm
descent converges almost surely to a global minimum when the objective function is convex or pseudoconvex, and otherwise converges almost surely to a local minimum
Stochastic_gradient_descent
Random measure in probability theory
converges to P(A) almost surely for fixed A. Similarly P n f {\displaystyle P_{n}f} converges to E f {\displaystyle \mathbb {E} f} almost surely for a fixed
Empirical_measure
Inequalities in probability theory
random variables. Suppose that | X i | ≤ M {\displaystyle |X_{i}|\leq M} almost surely, for all i . {\displaystyle i.} Then, for all positive t {\displaystyle
Bernstein inequalities (probability theory)
Bernstein_inequalities_(probability_theory)
Theorem in probability theory
k − X k − 1 | ≤ c k , {\displaystyle |X_{k}-X_{k-1}|\leq c_{k},\,} almost surely. Then for all positive integers N and all positive reals ϵ {\displaystyle
Azuma's_inequality
American governmental official (1896–1974)
enemy. Thus, in the absence of Strauss's action, the same decision almost surely would have been reached. In any case, when the decision was announced
Lewis_Strauss
Stochastic processes in mathematics
Doob's inequality states that if the sample paths of the martingale are almost-surely right-continuous, then: P [ sup 0 ≤ t ≤ T X t ≥ C ] ≤ E [ max ( X
Doob's_martingale_inequality
Special case in probability theory; introduces tail events
namely a tail event of independent σ-algebras, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring
Kolmogorov's_zero–one_law
Financial model
_{0}^{T}|\sum _{n=0}^{N}\pi _{n}(t)|\left[|r(t)|dt+dA(t)\right]<\infty } , almost surely, ∫ 0 T | ∑ n = 1 N π n ( t ) [ b n ( t ) + δ n ( t ) − r ( t ) ] | d
Brownian model of financial markets
Brownian_model_of_financial_markets
Indian self-rule activist (1856–1920)
other associates. According to Barbara and Thomas R. Metcalf, Tilak "almost surely concealed the identities of the perpetrators". Tilak was charged with
Bal_Gangadhar_Tilak
ALMOST SURELY
ALMOST SURELY
ALMOST SURELY
ALMOST SURELY
ALMOST SURELY
ALMOST SURELY
ALMOST SURELY
ALMOST SURELY
ALMOST SURELY