Advanced Mathematical Tools for Automatic Control Engineers: by Alexander S. Poznyak (Auth.)

By Alexander S. Poznyak (Auth.)

Content material:
Dedication

, Page ii
Copyright

, Page iv
Preface

, Pages xv-xix
Notations and Symbols

, Pages xxi-xxv
Chapter 1 - likelihood Space

, Pages 3-31
Chapter 2 - Random Variables

, Pages 33-45
Chapter three - Mathematical Expectation

, Pages 47-62
Chapter four - uncomplicated Probabilistic Inequalities

, Pages 63-81
Chapter five - attribute Functions

, Pages 83-99
Chapter 6 - Random Sequences

, Pages 103-131
Chapter 7 - Martingales

, Pages 133-173
Chapter eight - restrict Theorems as Invariant Laws

, Pages 175-236
Chapter nine - simple houses of continuing Time Processes

, Pages 239-261
Chapter 10 - Markov Processes

, Pages 263-286
Chapter eleven - Stochastic Integrals

, Pages 287-322
Chapter 12 - Stochastic Differential Equations

, Pages 323-354
Chapter thirteen - Parametric Identification

, Pages 357-416
Chapter 14 - Filtering, Prediction and Smoothing

, Pages 417-437
Chapter 15 - Stochastic Approximation

, Pages 439-470
Chapter sixteen - powerful Stochastic Control

, Pages 471-527
Bibliography

, Pages 529-533
Index

, Pages 535-538

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Additional info for Advanced Mathematical Tools for Automatic Control Engineers: Stochastic Techniques

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1. Let ξ1 , ξ2 . . be random variables. Then the following quantities are random variables too: 38 Advanced Mathematical Tools for Automatic Control Engineers: Volume 2 1. 14) 2. 15) n 3. 16) n 4. If {ξn (ω)} converges for all ω ∈ , then lim ξn (ω) is a random variable too. n→∞ Proof. 1. 14). 2. 15). 3. 16) notice that ω : lim sup ξn (ω) ≤ x = ω : lim sup ξn (ω) ≤ x n→∞ m≥n n ∞ = ω : inf ∞ ξn (ω) ≤ x sup ξn (ω) ≤ x = ω : n→∞ m≥n n=1 m=n ∈F 39 Random variables and ω : lim inf ξn (ω) ≤ x = ω : lim inf ξn (ω) ≤ x n n→∞ m≥n ∞ = ω : sup ∞ ξn (ω) ≤ x inf ξn (ω) ≤ x = ω : n→∞ m≥n ∈F n=1 m=n since by 2 inf ξn (ω) is a random variable, and again by 2 inf sup ξn (ω) ≤ x is a random m≥n n→∞m≥n variable too.

Corollary is proved. 5. 72) i=1 N Proof. ,N is a partition of we can conclude that i=1 Then using this identity we derive N χ (ω ∈ A) = χ (ω ∈ A) χ (ω ∈ Bi ) i=1 N χ (ω ∈ A) χ (ω ∈ Bi ) = i=1 N χ ((ω ∈ A) ∩ (ω ∈ Bi )) χ (ω ∈ Bi ) = i=1 N χ ((ω ∈ A) ∩ (ω ∈ Bi )) = i=1 χ (ω ∈ Bi ) = 1. 31 Probability space Here we have used χ (ω ∈ A) χ (ω ∈ Bi ) = χ ((ω ∈ A) ∩ (ω ∈ Bi )) χ (ω ∈ Bi ) = χ ((ω ∈ A) ∩ (ω ∈ Bi )) N for any sets A, Bi ∈ F. This exactly means that P {A} = P {A ∩ Bi }. 72). The corollary is proven.

The next definition is the central one in this chapter. 4. 6) k=1 Intuitively, a random variable is a quantity that is measured in connection with a random experiment: if ( , F, P) is a probability space and the outcome of the experiment corresponds to the point ω ∈ , a measuring process is carried out to obtain a number ξ(ω). Thus, ξ = ξ(ω) is a function from the sample space to the reals (or extended reals including ±∞) R N . 2. If we are interested in a random variable ξ defined on a given probability space, we generally want to know the probability of all events involving ξ .

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