By Petr Mandl

ISBN-10: 0387041427

ISBN-13: 9780387041421

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**Additional info for Analytical treatment of one-dimensional Markov processes**

**Example text**

505]) the sequence {Zt (τ ) = eτ Yt −ΛD (τ ) Y t , t ≥ 1} with Z0 (τ ) = 1 forms a non-negative supermartingale. From the above inequality, it follows that P 1 M M Xi − µ ≥ , M ≥ n i=1 ≤ P (eτ YM −ΛD (τ ) ≤P ≤ Y M ≥ eτ n sup Zt (τ ) ≥ eτ n −nΛD (τ )D2 −nΛD (τ )D ) 2 0≤t≤L E[Z0 (τ )] τ n e −nΛD (τ )D2 = e−n(τ −ΛD (τ )D2 ) by maximal inequality for supermartingales [133] . 2 Pursuit Learning Automata Sampling 45 By using a similar argument, we can also show that P 1 M M Xi − µ ≤ − , M ≥ n ≤ e−n(τ −ΛD (τ )D2 ) .

5. 25) where the operator in deﬁning a ˆ ∈ arg maxa {Nai (x)} remains a maximization operation. With K = 0 (no ﬁxed order cost), the optimal order policy is easily solvable without dynamic programming, because the periods are decoupled, and the problem reduces to solving a single-period inventory optimization problem. In case (i), the optimal policy follows a threshold rule, in which an order is placed if the inventory is below a certain level; otherwise, no order is placed. , an order will always be placed).

We analyze the ﬁnite-time behavior of the PLA sampling algorithm, providing a bound on the probability that a given initial state takes the optimal action, and a bound on the probability that the diﬀerence between the optimal value and the estimate of it exceeds a given error. Similar to the UCB algorithm, the PLA sampling algorithm constructs a sampled tree in a recursive manner to estimate the optimal value at an initial state and incorporates an adaptive sampling mechanism for selecting which action to simulate at each branch in the tree.

### Analytical treatment of one-dimensional Markov processes by Petr Mandl

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