Lyapunov condition
E683054
condition for central limit theorem
condition in probability theory
moment condition
sufficient condition
The Lyapunov condition is a sufficient moment condition on sums of independent random variables that guarantees convergence in distribution to a normal law in central limit theorems.
Statements (32)
| Predicate | Object |
|---|---|
| instanceOf |
condition for central limit theorem
ⓘ
condition in probability theory ⓘ moment condition ⓘ sufficient condition ⓘ |
| appliesTo |
sums of independent random variables
ⓘ
triangular arrays of independent random variables ⓘ |
| assumes |
independence of summands
ⓘ
non-degenerate variance of the sum ⓘ |
| comparedTo | Lindeberg–Feller condition NERFINISHED ⓘ |
| domain |
asymptotic distribution theory
ⓘ
limit theorems ⓘ |
| ensures |
convergence in distribution to a normal law
ⓘ
convergence of characteristic functions to that of a normal distribution ⓘ negligibility of large individual summands ⓘ |
| field |
mathematical statistics
ⓘ
probability theory ⓘ |
| guarantees | asymptotic normality of standardized sums ⓘ |
| historicalContext | introduced in early 20th century ⓘ |
| implies | Lindeberg condition ⓘ |
| involves |
Lyapunov fraction
NERFINISHED
ⓘ
normalization by variance of the sum ⓘ |
| is | a classical form of central limit theorem hypothesis ⓘ |
| namedAfter | Aleksandr Lyapunov NERFINISHED ⓘ |
| relatedTo | Berry–Esseen bounds NERFINISHED ⓘ |
| requires |
existence of moments of order greater than 2
ⓘ
finite (2 + δ)-th absolute moments for some δ > 0 ⓘ |
| strongerThan | Lindeberg condition NERFINISHED ⓘ |
| typeOf | sufficient central limit theorem condition ⓘ |
| usedFor |
controlling tail behavior of summands
ⓘ
proving central limit theorems for non-identically distributed variables ⓘ |
| usedIn |
Lyapunov central limit theorem
NERFINISHED
ⓘ
central limit theorem NERFINISHED ⓘ |
Referenced by (1)
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