Khinchin's law of the iterated logarithm

E378993

law of the iterated logarithm probability theorem result in probability theory

Khinchin's law of the iterated logarithm is a fundamental result in probability theory that precisely characterizes the almost-sure fluctuations of partial sums of independent random variables on the scale of the square root of twice the product of their variance and the iterated logarithm of the sample size.

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All labels observed (2)

Label	Occurrences
Khinchin's law of the iterated logarithm canonical	3
Hartman–Wintner law of the iterated logarithm	1

Statements (46)

Predicate	Object
instanceOf	law of the iterated logarithm ⓘ probability theorem ⓘ result in probability theory ⓘ
appliesTo	independent identically distributed random variables with zero mean ⓘ random variables with finite nonzero variance ⓘ
assumes	finite second moment ⓘ identical distribution of the summands ⓘ independence of the summands ⓘ
characterizes	almost-sure fluctuations of partial sums of independent random variables ⓘ
comparesWith	central limit theorem scaling sqrt(n) ⓘ strong law of large numbers scaling n ⓘ
concerns	almost sure convergence properties ⓘ partial sums of random variables ⓘ
describes	asymptotic behavior of normalized partial sums ⓘ limiting envelope of normalized random walk paths ⓘ
field	probability theory ⓘ stochastic processes ⓘ
gives	exact liminf behavior of normalized partial sums ⓘ exact limsup behavior of normalized partial sums ⓘ
givesInformationOn	maximal fluctuations of partial sums ⓘ oscillatory behavior of sums around zero ⓘ
historicalPeriod	early 20th century ⓘ
implies	liminf of S_n divided by sqrt(2 σ² n log log n) equals -1 almost surely ⓘ limsup of S_n divided by sqrt(2 σ² n log log n) equals 1 almost surely ⓘ
isFormulatedFor	partial sums S_n = X_1 + ... + X_n ⓘ
isRelatedTo	Brownian motion ⓘ Donsker's invariance principle ⓘ Khinchin's law of the iterated logarithm self-linksurface differs ⓘ surface form: Hartman–Wintner law of the iterated logarithm Kolmogorov's law of the iterated logarithm ⓘ functional law of the iterated logarithm ⓘ
isSpecialCaseOf	Kolmogorov's law of the iterated logarithm ⓘ
isUsedIn	asymptotic analysis of stochastic processes ⓘ empirical process theory ⓘ limit theory of random walks ⓘ probabilistic number theory ⓘ statistics of extremes of partial sums ⓘ
namedAfter	Aleksandr Khinchin ⓘ
normalizationInvolves	square root of 2 σ² n log log n ⓘ
refines	central limit theorem ⓘ law of large numbers ⓘ surface form: strong law of large numbers
requires	nondegenerate variance ⓘ
scaleOfFluctuations	square root of twice the variance times the iterated logarithm of sample size ⓘ
strengthens	information provided by the central limit theorem about fluctuations ⓘ information provided by the strong law of large numbers about convergence ⓘ
typeOfLimit	almost sure limit theorem ⓘ
usesFunction	iterated logarithm log log n ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Aleksandr Khinchin → notableWork → Khinchin's law of the iterated logarithm ⓘ

Aleksandr Khinchin → notableIdea → Khinchin's law of the iterated logarithm ⓘ

Khinchin → notableFor → Khinchin's law of the iterated logarithm ⓘ

subject surface form: Aleksandr Khinchin

Khinchin's law of the iterated logarithm → isRelatedTo → Khinchin's law of the iterated logarithm self-linksurface differs ⓘ

this entity surface form: Hartman–Wintner law of the iterated logarithm