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.

Sample Behavior and Laws of Large Numbers for Gaussian Random Elements

2003 ◽  
Vol 10 (4) ◽  
pp. 637-676
Author(s):  
Z. Ergemlidze ◽  
A. Shangua ◽  
V. Tarieladze
Keyword(s):  
Banach Space ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Random Elements ◽  
Vector Valued

Abstract Criteria for almost sure boundedness and convergence to zero almost surely of Banach space valued independent Gaussian random elements are found. The obtained statements can be viewed as vector-valued versions of the corresponding results due to N. Vakhania. Moreover, from the obtained statements a strong law of large numbers is derived in the form of Yu. V. Prokhorov.

Explicit Stable Strong Laws of Large Numbers

1994 ◽  
Vol 44 (3-4) ◽  
pp. 141-150 ◽  
Author(s):  
André Adler
Keyword(s):  
Law Of Large Numbers ◽  
Random Variables ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Interesting Class

In this article it is shown, through a very interesting class of random variables, how one may go about explicitly obtaining constants in order to obtain a stable strong law of large numbers. The question at hand is, not when we can find constants an and bn so that our sequence of i. i.d. random variables obeys this type of strong law of large numbers, but how one goes about constructing these constants so that [Formula: see text] almost surely, even though { X, Xn} are i.i.d. with either [Formula: see text] There are three possible cases. We exhibit all three via a particular family of random variables.

scholarly journals Strong laws of large numbers for arrays of row-wise exchangeable random elements

1985 ◽  
Vol 8 (1) ◽  
pp. 135-144 ◽  
Author(s):  
Robert Lee Taylor ◽  
Ronald Frank Patterson
Keyword(s):  
Banach Space ◽  
Almost Sure Convergence ◽  
Kernel Density ◽  
Triangular Array ◽  
Density Estimates ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Kernel Density Estimates ◽  
Large Numbers ◽  
Random Elements

Let{Xnk,1≤k≤n,n≤1}be a triangular array of row-wise exchangeable random elements in a separable Banach space. The almost sure convergence ofn−1/p∑k=1nXnk,1≤p<2, is obtained under varying moment and distribution conditions on the random elements. In particular, strong laws of large numbers follow for triangular arrays of random elements in(Rademacher) typepseparable Banach spaces. Consistency of the kernel density estimates can be obtained in this setting.

scholarly journals On strong laws of large numbers for arrays of rowwise independent random elements

1993 ◽  
Vol 16 (3) ◽  
pp. 587-591 ◽  
Author(s):  
Abolghassem Bozorgnia ◽  
Ronald Frank Patterson ◽  
Robert Lee Taylor
Keyword(s):  
Banach Space ◽  
Density Estimation ◽  
Separable Banach Space ◽  
Complete Convergence ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Random Elements ◽  
Rowwise Independent

Let{Xnk}be an array of rowwise independent random elements in a separable Banach space of typer,1≤r≤2. Complete convergence ofn1/p∑k=1nXnkto0,0<p<r≤2is obtained whensup1≤k≤nE ‖Xnk‖v=O(nα),α≥0withv(1p−1r)>α+1. An application to density estimation is also given.

scholarly journals On Chung's strong law of large numbers in general Banach spaces

1988 ◽  
Vol 37 (1) ◽  
pp. 93-100 ◽  
Author(s):  
Bong Dae Choi ◽  
Soo Hak Sung
Keyword(s):  
Banach Spaces ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Real Numbers ◽  
Strong Law ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers

Let { Xn, n ≥ 1 } be a sequence of independent Banach valued random variables and { an, n, ≥ 1 } a sequence of real numbers such that 0 < an ↑ ∞. It is shown that, under the assumption with some restrictions on φ, Sn/an → 0 a.s. if and only if Sn/an → 0 in probability if and only if Sn/an → 0 in L1. From this result several known strong laws of large numbers in Banach spaces are easily derived.

scholarly journals On the strong law for arrays and for the bootstrap mean and variance

1997 ◽  
Vol 20 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Tien-Chung Hu ◽  
R. L. Taylor
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Interesting Application ◽  
Strong Law ◽  
Moment Conditions ◽  
Laws Of Large Numbers ◽  
Strong Laws ◽  
Large Numbers ◽  
Mean And Variance ◽  
Rowwise Independent

Chung type strong laws of large numbers are obtained for arrays of rowwise independent random variables under various moment conditions. An interesting application of these results is the consistency of the bootstrap mean and variance.

Sign in / Sign up

Export Citation Format

Share Document