СТАТИСТИЧНИЙ АНАЛІЗ ЛОКАЛЬНИХ ДІЛЯНОК БІТОВИХ ПОСЛІДОВНОСТЕЙ
The joint distributions of the number of 2-chains and the number of 3-chains of a fixed form of a random (0, 1)-sequence, which allow a statistical analysis of local sections of this sequence, were examined. Two theorems are formulated and proved. Consider s-chains of the form t t* , * t1t *,&nb...
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| Datum: | 2025 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
V.M. Glushkov Institute of Cybernetics of NAS of Ukraine
2025
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| Online Zugang: | https://jais.net.ua/index.php/files/article/view/683 |
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| Назва журналу: | Problems of Control and Informatics |
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Problems of Control and Informatics| Zusammenfassung: | The joint distributions of the number of 2-chains and the number of 3-chains of a fixed form of a random (0, 1)-sequence, which allow a statistical analysis of local sections of this sequence, were examined. Two theorems are formulated and proved. Consider s-chains of the form t t* , * t1t *, t 0 t* , t1t1* , t11t1* ( t1t1*, t1t, t 0t, t t t, t t* t) which appeared in random bit sequence of fixed length in Theorems 1, 2. For these s-chains, explicit expressions for the joint distributions of such events were established: {h(t t* ) = k1, h(t1t* ) + h(t 0t* ) = k2}, {h(t1 t1* ) = k1, h(t1 t t1* ) = k2}, {h(t t *) = k1, h(t1t* ) = k2, h(t 0t* ) = k3}, ({h(t1 t1* ) = k1, h(t1t) + h(t 0t) = k2}, {h(t1 t1* ) = k1, h(t t t) = k2}, {h(t1 t1* ) = k1, h(t t *t) = k2}, {h(t1 t1* ) = k1, h(t t t) = k2, h(t t *t) = k3}), where h(t1 t2 ... ts ) is the number of s-chains of the form t1 t2 ... ts in the initial n-dimensional (0, 1)-sequence; k1 , k2 and k3 are suitable non-negative integers. One of the main assumptions of each theorem is that zeros and ones in a bit sequence are independent identically distributed
random variables. The proofs of the formulas for the distributions of these events are based on counting the number of corresponding conductive events, provided that the (0, 1)-sequence contains a fixed number of zeros and ones. As examples of the use of explicit expressions of joint distributions, tables that contain the probabilities of the above events for a random (0, 1)-sequence of length n, n = 20, and some values of the parameters k1 , k2 and k3 under the assumption that zeros and units appear equally likely are given. For illustrative purposes, some of the tables are illustrated by bubble chart. The established formulas may be of interest for tasks like testing local sections formed at the output of pseudorandom number generators. Also, they may be suitable for some tasks of protecting information from unauthorized access, as well as in other areas where it becomes necessary to analyze bit sequences.
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