By Alexander S. Poznyak (Auth.)

ISBN-10: 0080446736

ISBN-13: 9780080446738

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 - simple 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 - uncomplicated homes 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 - strong 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**

**Example text**

N) and Ai := {ω : ω ∈ [i − 1, i)}. Then the random variable ξ(ω) is a simple random variable (see Fig. 3). 2 Transformation of distributions This section describes some helpful rules for the calculation of distribution functions of random variables in the case of their functional transformations. 1. Let ξ = ξ(ω) be a random variable, defined on ( , F, P), and φ = φ(x) be a Borel function. Then the function η(ω) = φ(ξ(ω)) is a random function defined on the same probability space ( , F, P).

X N ) with the following properties: 1. for any a, b ∈ R N a,b F(x) (bi ≥ ai , i = 1, . . , N ) := F(b1 , b2 , . . , b N ) − F(a1 , a2 , . . , a N ) = F(b) − F(a) ≥ 0 2. , F(x (k) ) ↓ F(x) i f x (k) ↓ x and has a bounded limit on left; 3. F(+∞, +∞, . . , +∞) = 1 and lim F(x) = 0 x↓y if at least one of coordinate yi of a vector y ∈ R N is equal to (−∞). 4 also takes place. 6. For each distribution function F = F(x), x ∈ R N there exists a unique probability measure P on R N , B(R N ) such that for any a, b : −∞ ≤ ai < bi ≤ ∞, i = 1, .

3. If A ⊂ and B ⊂ A and B c , Ac and B, Ac and B c . are independent, then so are Proof. One has P A ∩ B c = P {A} − P {A ∩ B} = P {A} − P {A} P {B} = P {A} (1 − P {B}) = P {A} P B c Analogously, P Ac ∩ B = P {B} − P {A ∩ B} = P {B} − P {A} P {B} = P {B} (1 − P {A}) = P {B} P Ac and P Ac ∩ B c = P Ac − P Ac ∩ B = P Ac − P Ac P {B} = P Ac (1 − P {B}) = P Ac P B c which completes the proof. 7. If {Ak , k = 1, . . 63) i=1 • n P n Ai P{Ai } ≥ 1 − exp − i=1 i=1 Proof. It can be done by induction. 63) for any integer n.

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

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