On the transfer theorems for observed and
unobserved random variables
Abstract:
We characterize the possible weak limits of \[\sum _{i=1}^n \epsilon _iX_i/k_n\]
for a sequence \(\{X_n, n\geq 1\}\) of independent random variables
and a sequence \(\{\epsilon _n, n\geq
1\}\) of indicator random variables (\(P[\epsilon _n\in\{0,1\}]=1\)
for \(n\geq 1\)) and a non-random normalizing sequence \(\{k_n, n\geq 1\}\)
of positive reals. We consider two cases: when \(\{X_n, n\geq 1\}\) and \(\{\epsilon _n,
n\geq 1\}\) are independent or dependent.
In the first case we obtain results generalizing transfer theorems, whereas in the other case, only a partial characterization was possible.
2010 AMS Mathematics Subject Classification: Primary 60F05.
Keywords and phrases: transfer theorem, weak limits, sums of observed random
variables.
