We give conditions for the convergence in probability to a constant of suitably
normed variables derived from $T_n=
\sum_{k=1}^nb_k^{(n)}X_k$, where $X_1, X_2,\ldots$ is a sequence of nonnegative,
independent and identically distributed random variables with infinite mean and
distribution function
$F$, and $\{b_k^{(n)}\}$ are some positive constants with $\sum_{n=1}^nb_k^{(n)}=n$.
It is also assumed that
$\int_0^u(1-F(u))\hbox{du}$ is a slowly varying function. The method used involves
truncations both of large values of the variables $\{X_n\}$ and of small values
of the weights $\{b_k^{(n)}\}$.