Limit theorems for the number of successes in random binary sequences with random embeddings

The sequence of $ n $ random $ (0,1) $-variables $ X_1,\,\ldots \, , \, X_n $ is considered, with $ \theta_n $ of these variables distributed equiprobable and the others take the value 1 with probability $ p $ ($ 0 < p < 1, p \neq 1/2 $), $\theta_n $ is a random variable taking values $ 0,\,1,\,\ldots ,\,n $). On the assumption that $ n \to \infty $ and under certain conditions imposed on $ p,\theta_n $ and $ X_k,\,k = 1,\ldots, n, $ several limit theorems for the sum $ S_n = \sum_{k=1}^n X_k $. The results are of interest in connection with steganography and statistical analysis of sequences produced by random number generators.