The Vernam cipher is robust to small deviations from randomness

The Vernam cipher (or one-time pad) has played an important rule in cryptography because it is a perfect secrecy system. For example, if an English text (presented in binary system) X1X2... is enciphered according to the formula Zi = (Xi + Yi) mod 2, where Y1Y2... is a key sequence generated by the Bernoulli source with equal probabilities of 0 and 1, anyone who knows Z1Z2... has no information about X1X2... without the knowledge of the key Y1Y2.... (The best strategy is to guess X1X2... not paying attention to Z1Z2....) But what should one say about secrecy of an analogous method where the key sequence Y1Y2... is generated by the Bernoulli source with a small bias, say, P (0) = 0.49, P (1) = 0.51? To the best of our knowledge, there are no theoretical estimates for the secrecy of such a system, as well as for the general case where X1X2... (the plaintext) and key sequence are described by stationary ergodic processes. We consider the runningkey ciphers where the plaintext and the key are generated by stationary