Xll
Preface
instructional issue s ar e wel l understood , an d mos t expert s agre e o n wha t
should b e taught .
Giving a one-semeste r introductio n t o graduat e probabilit y necessaril y
involves makin g concessions . Min e for m th e content s o f thi s book : N o men -
tion i s mad e o f Kolmogorov' s theor y o f rando m series ; Levy' s continuit y
theorem o f characteristi c function s i s sadl y omitted ; Marko v chain s ar e no t
treated a t all ; an d th e constructio n o f Brownia n motio n i s Fourier-analyti c
rather tha n "probabilistic. "
Tha t i s no t t o sa y tha t ther e i s littl e coverag e o f th e theor y o f stochasti c
processes. Fo r example , include d yo u wil l fin d a n introductio n t o Ito' s sto -
chastic calculu s an d it s connection s t o ellipti c partia l differentia l equations .
This topi c ma y see m ambitious , an d i t probabl y i s fo r som e readers . How -
ever, m y experienc e i n teachin g thi s materia l ha s bee n tha t th e reade r wh o
knows som e measur e theor y ca n cove r th e boo k u p t o an d includin g th e las t
chapter i n a singl e semester . Thos e wh o wis h t o lear n measur e theor y fro m
this boo k woul d probabl y ai m t o cove r les s stochasti c processes .
Teachin g R e c o m m e n d a t i o n s . I n m y ow n lecture s I ofte n begi n wit h
Chapter 2 an d prov e th e D e Moivre-Laplac e centra l limi t theore m i n detail .
Then, I spen d tw o o r thre e week s goin g ove r basi c result s i n analysi s [Chap -
ters 3 throug h 5] . Onl y a handfu l o f th e sai d result s ar e actuall y proved .
Without exception , on e o f the m i s Caratheodory' s monoton e clas s theore m
(p. 30) . Th e fundamenta l notio n o f independenc e i s introduced , an d a num -
ber o f importan t example s ar e worke d out . Amon g the m ar e th e wea k an d
the stron g law s o f larg e number s [Chapte r 6] , respectivel y du e t o A . Ya .
Khintchine an d A . N . Kolmogorov . Nex t follo w element s o f harmoni c anal -
ysis an d th e centra l limi t theore m [Chapte r 7] . A majorit y o f th e subsequen t
lectures concer n J . L . Doob' s theor y o f martingale s (1940) an d it s variou s
applications [Chapte r 8] . Afte r martingales , ther e ma y b e enoug h tim e lef t
to introduc e Brownia n motio n [Chapte r 9] , construc t stochasti c integrals ,
and deduc e a strikin g computation , du e t o Chun g (1947), o f th e distribu -
tion o f th e exi t tim e fro m [—1,1o ] f Brownia n motio n (p . 197). I f a t al l
possible, th e latte r topi c shoul d no t b e missed .
My persona l teachin g philosoph y i s t o showcas e th e bi g idea s o f proba -
bility b y derivin g ver y few , bu t central , theorems . Frangoi s Mari e Aroue t
[Voltaire] onc e wrot e tha t "th e ar t o f bein g a bor e i s t o tel l everything. "
Viewed i n thi s light , a chie f ai m o f thi s boo k i s t o no t bore .
I woul d lik e t o leav e th e reade r wit h on e piec e o f advic e o n ho w t o bes t
use thi s book . Rea d i t thoughtfully , an d wit h pe n an d paper .
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