## Abstract

The general problem under investigation is to understand how the complexity of a system which has been adapted to its random environment affects the level of randomness of its output (which is a function of its random input). In this paper, we consider a specific instance of this problem in which a deterministic finite-state decision system operates in a random environment that is modeled by a binary Markov chain. The system interacts with it by trying to match states of inactivity (represented by 0). Matching means that the system selects the (t + 1)th bit from the Markov chain whenever it predicts at time t that the environment will take a 0 value. The actual value at time t + 1 may be 0 or 1 thus the selected sequence of bits (which forms the system's output) may have both binary values. To try to predict well, the system's decision function is inferred based on a sample of the random environment. We are interested in assessing how non-random the output sequence may be. To do that, we apply the adapted system on a second random sample of the environment and derive an upper bound on the deviation between the average number of 1 bit in the output sequence and the probability of a 1. The bound shows that the complexity of the system has a direct effect on this deviation and hence on how non-random the output sequence may be. The bound takes the form of O(√(2^{k}/n)) where 2^{k} is the complexity of the system and n is the length of the second sample.

Original language | English |
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Pages (from-to) | 528-563 |

Number of pages | 36 |

Journal | Probability in the Engineering and Informational Sciences |

Volume | 33 |

Issue number | 4 |

DOIs | |

State | Published - 1 Oct 2019 |

Externally published | Yes |

## Keywords

- Markov chain
- frequency instability
- prediction of random binary sequence
- subsequence selection