Simple Derivation of the Dissociation Formula

CRITICISM

AND

BIBLIOGRAPHY

A. V. S k o r o k h o d ELEMENTS RANDOM

OF

PROBABILITY

THEORY

AND

PROCESSES*

Reviewedby V. S. K o r o l y u k

T h i s textbook for students of m e c h a n i c o m a t h e m a t i c a [ and applied m a t h e m a t i c s faculties contains a c o n c i s e p r e s e n t a t i o n of the b a s i c facts of probability theory and a d e s c r i p t i o n of the m a i n c l a s s e s of r a n d o m p r o c e s s e s . The author o r g a n i z e d his m a t e r i a l in an original way, p r e s e n t i n g r e s u l t s usually r e g a r d e d as b e l o n g ing to p r o b a b i l i t y theory c o n c u r r e n t l y with r e s u l t s which a r e traditionaliy included in the theory of stochastic p r o c e s s e s . T h i s m a k e s it p o s s i b l e to f a m i l i a r i z e the students i m m e d i a t e l y with the p r a c t i c a l l y i m p o r t a n t c l a s s e s of r a n d o m p r o c e s s e s and leads them to the r e a l i z a t i o n of the n e c e s s i t y of examining the basic m a t h e m a t i c a l concepts of the theory of r a n d o m p r o c e s s e s . Each section is a c c o m p a n i e d by p r o b l e m s ; the solution of s o m e of them involves proving s o m e well-known t h e o r e m s . In the text which follows t h e m , the r e s u l t s obtained while solving the p r o b l e m a r e used on a p a r with the p r e v i o u s l y obtained t h e o r e m s . A total of m o r e than 370 p r o b l e m s a r e mentioned in the book. T h e book c o n s i s t s of eight c h a p t e r s . C h a p t e r 1 introduces the basic c o n c e p t s , such as a l g e b r a s and a a l g e b r a s of events, p r o b a b i l i t y , r a n d o m v a r i a b l e s and t h e i r distributions, expectations, conditional p r o b a b i l i t i e s , and conditional expectations. C h a p t e r 2 is devoted to sequences of independent r a n d o m v a r i a b l e s and events. H e r e , the subjects d i s c u s s e d include the z e r o - o n e law, the Canteili t h e o r e m , binomial distributions with a s y m p t o t i c f o r m u l a s , the P o i s s o n p r o c e s s , the c o n v e r g e n c e of s e r i e s of independent r a n d o m v a r i a b l e s , the s t r o n g law of large n u m b e r s , and m a r t i n g a l e s . It is a p p a r e n t that the topics c o n c e r n , to a large extent, the theory of r a n d o m p r o c e s s e s . C h a p t e r 3 d i s c u s s e s renewal p r o c e s s e s , r a n d o m w a l k s , and the g e n e r a l i z e d P o i s s o n p r o c e s s . A s is well known, the c e n t r a l limit t h e o r e m of probability theory is intimately connected with continuous p r o c e s s e s with independent i n c r e m e n t s , i.e., with Brownian motion. T h e s e p r o b l e m s a r e d i s c u s s e d in Chap. 4. We note that h e r e local t h e o r e m s a r e introduced with a r i g o r which was p r e v i o u s l y lacking in text books. C h a p t e r 5, which is devoted to infinitely divisible distributions, contains a discussion of g e n e r a l limit t h e o r e m s on s u m s of independent r a n d o m v a r i a b l e s , and of s t o c h a s t i c a l l y continuous p r o c e s s e s with independent i n c r e m e n t s . T h e p r e s e n t a t i o n of these p r o b l e m s in p a r a l l e l allows one to a c q u i r e a d e e p e r insight into each. T h e last t h r e e c h a p t e r s a r e devoted to t h r e e c l a s s e s of random p r o c e s s e s : Markov chains with continuous t i m e , branching p r o c e s s e s , and s t a t i o n a r y p r o c e s s e s . T h e book is w r i t t e n in a v e r y condensed f o r m , and at the s a m e t i m e it illumirmtes p r o b l e m s pertaining to the u n i v e r s i t y syllabus. The p r e s e n t a t i o n is thoroughly consistent. A p a r t of the contents of the book can be used in c o u r s e s for s p e c i a l i s t s (Chaps. 5-8). The only drawback is the e x c e s s i v e l y c o m p r e s s e d p r e s e n t a t i o n of the basic concepts in Chap. 1. The Ukrainian edition (2000 copies) is a l r e a d y out of print. In a f u r t h e r e d i tion, Chap. 1 should be amplified. It may be that in o r d e r to keep the book within the s a m e limits of length, the s p e c i a l p a r t (Chaps. 5-8) should be curtailed.

*Naukova Dumka, Kiev (1977). T r a n s l a t e d f r o m Ukrainskti M a t e m a t i c h e s k i i Zhurnai, VoI. 30, No. 1, p. 139, J a n u a r y - F e b r u a r y , 1978. 0041-5995/78/3001- 0109507,50 9 1978 Plenum Publishing C o r p o r a t i o n

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