Let \((X_n)\) be a sequence of independent random variables taking values in a topological semigroup \(S\). Let the probability measure \(\mu_n\) be the distribution of \(X_n\). The paper aims at determining conditions under which the non-homogeneous random walk \(X_{k + 1} X_{k + 2} \cdots X_n\) converges in distribution for all \(k \geq 0\). In terms of the sequence \((\mu_n)\) this amounts to finding conditions for the convolutions \(\mu_{k + 1} * \mu_{k + 2} * \cdots * \mu_n\) to converge weakly for all \(k\). -- From the abstract: `` \dots a measure \(\lambda\) on \(S\) is called a tail limit of \((\mu_n)\) if, for some subsequence of integers \((n_i)\), \(\mu_{k,n_i} = \mu_{k + 1} * \cdots * \mu_{n_i}\) converges weakly to \(\nu_k\) for all \(k\) and \(\lambda\) is a weak limit point of the sequence \((\nu_k)\). The main theorem of this paper characterizes the supports of the tail limits on compact completely simple semigroups. Some applications of the theorem and various open problems are discussed''.