Self-adjusting stochastic processes |
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Authors: | Emilio Gagliardo Clifford Kottman |
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Institution: | Istituto Matematico, Università, Pavia 27100, Italia;Defense Mapping Agency, Washington, DC 20315, U.S.A. |
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Abstract: | Let At(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that At+1(h, k) ? At(h, k). If the self-adjustment (due to transition h → kx) is At + 1(h, j) = λAt(h, j) + (1 ? λ)δjk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A0(i, j) < 1 for all i, j and if the self-adjustment satisfies At + 1(h, k) = ?(At(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim At does not exist. |
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Keywords: | Adaptive process transition matrix |
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