Abstract: Word embeddings generated by neural network methods such as word2vec (W2V) are well known to exhibit seemingly linear behaviour, e.g. the embeddings of analogy ``woman is to queen as man is to king'' approximately describe a parallelogram. This property is particularly intriguing since the embeddings are not trained to achieve it. Several explanations have been proposed, but each introduces assumptions that do not hold in practice. We derive a probabilistically grounded definition of paraphrasing that we re-interpret as word transformation, a mathematical description of ``$w_x$ is to $w_y$''. From these concepts we prove existence of linear relationship between W2V-type embeddings that underlie the analogical phenomenon, identifying explicit error terms.