Summary: For a sequence of arbitrary real-valued random variables \(\{X_n, n\geq 1\}\) and a sequence of functions \(\{\varphi_n(x), n\geq 1\}\), we study strong limit theorems of the random sequence \(\{\varphi_n (X_n), n\geq 1\}\). The theorems generalize some classical limit theorems for the sequence of independent random variables.