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(7)
Classical Limit
Independent Random Variables
Law of Large Numbers
Limit Theorem
Moment Condition
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The 1971 Rietz Lecture Sums of Independent Random VariablesWithout Moment Conditions
The 1971 Rietz Lecture Sums of Independent Random VariablesWithout Moment Conditions,10.1214/aoms/1177692541,The Annals of Mathematical Statistics,H
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The 1971 Rietz Lecture Sums of Independent Random VariablesWithout Moment Conditions
(
Citations: 14
)
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Harry Kesten
Analogues of
classical limit
laws for sums of
independent random variables
(central limit theorem, laws of large numbers and law of the iterated logarithm) are discussed. We stress results which go through without moment or smoothness assumptions on the underlying distributions. These include (i) estimates for the spread of the distribution of $S_n = \sum_1^nX_i$ in terms of concentration functions (LevyRogozin inequality), (ii) comparison of the distribution of $S_n$ on different intervals (ratio limit theorems and Spitzer's theorem for the existence of the potential kernel for recurrent random walk), (iii) study of the set of accumulation points of $S_n/\Upsilon(n)$ for suitable $\Upsilon (n) \uparrow \infty$. Only the following parallel to the law of the iterated logarithm is new: If $X_1, X_2, \cdots$ are
independent random variables
all with distribution $F, S_n = \sum_1^nX_1, m_n = \operatorname{med} (S_n)$, then there exists a sequence $\{\Upsilon (n)\}$ such that $\Upsilon (n) \rightarrow \infty$ and $ \infty < \lim \inf (S_n  m_n)/\Upsilon (n) < \lim \sup (S_n  m_n)/\Upsilon (n) < \infty$ w.p. 1, if and only if $F$ belongs to the domain of partial attraction of the normal law.
Journal:
The Annals of Mathematical Statistics
, vol. 43, no. 1972, pp. 701732, 1972
DOI:
10.1214/aoms/1177692541
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Citations
(14)
LIL behavior for weakly dependent random variables in Banach spaces
(
Citations: 1
)
K.A. Fu
Journal:
Acta Mathematica Hungarica  ACTA MATH HUNG
, vol. 128, no. 4, pp. 315327, 2010
Small time twosided LIL behavior for Lévy processes at zero
(
Citations: 4
)
Mladen Savov
Journal:
Probability Theory and Related Fields  PROBAB THEORY RELAT FIELD
, vol. 144, no. 1, pp. 7998, 2009
A general LIL for trimmed sums of random fields in banach spaces
(
Citations: 1
)
K.A. Fu
,
L.X. Zhang
Journal:
Acta Mathematica Hungarica  ACTA MATH HUNG
, vol. 122, no. 1, pp. 91103, 2009
A LIL for independent nonidentically distributed random variables in Banach space and its applications
(
Citations: 1
)
Weidong Liu
,
Keang Fu
,
Lixin Zhang
Journal:
Science Chinamathematics  SCI CHINAMATH
, vol. 51, no. 2, pp. 219232, 2008
A Nonclassical Law of the Iterated Logarithm for Functions of Positively Associated Random Variables
(
Citations: 1
)
JianFeng Wang
,
LiXin Zhang
Journal:
Metrika
, vol. 64, no. 3, pp. 361378, 2006