Convergence of Series of Scalar- and Vector-Valued Random Variables and a Subsequence Principle in L 2

1987 ◽  
Vol 301 (1) ◽  
pp. 375
Author(s):  
S. J. Dilworth
Keyword(s):  
Random Variables ◽  
Convergence Of Series ◽  
Vector Valued ◽  
Subsequence Principle

Subsequence principles for vector-valued random variables

1979 ◽  
Vol 86 (2) ◽  
pp. 301-312 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Principal Result ◽  
Moment Condition ◽  
Special Cases ◽  
Cesaro Averages ◽  
Image Position ◽  
Vector Valued ◽  
Subsequence Principle ◽  
Independent Identically Distributed

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such thatalmost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

scholarly journals On Chung-Teicher type strong law for arrays of vector-valued random variables

2004 ◽  
Vol 2004 (9) ◽  
pp. 443-458
Author(s):  
Anna Kuczmaszewska
Keyword(s):  
Banach Space ◽  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Random Elements ◽  
Vector Valued

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach spaceℬ. The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These conditions are expressed in terms of convergence of specific series ando(1)requirements on specific weighted row-wise sums. Moreover, there are not any conditions assumed on the geometry of the underlying Banach space.

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