Abstract:

We characterize the possible weak limits of $$\sum _{i=1}^n{ϵ}_i{X}_i/k_n$$ for a sequence $\{X_n,n\ge 1\}$ of independent random variables and a sequence $\{{ϵ}_n,n\ge 1\}$ of indicator random variables ($P[{ϵ}_n\in \{0,1\}]=1$ for $n\ge 1$) and a non-random normalizing sequence $\{k_n,n\ge 1\}$ of positive reals. We consider two cases: when $\{X_n,n\ge 1\}$ and $\{{ϵ}_n,n\ge 1\}$ 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.