A First Look at Rigorous Probability Theory by Jeffrey S. Rosenthal

Preface to the First Edition     vii
Preface to the Second Edition     xi
The need for measure theory     1
Various kinds of random variables     1
The uniform distribution and non-measurable sets     2
Exercises     4
Section summary     5
Probability triples     7
Basic definition     7
Constructing probability triples     8
The Extension Theorem     10
Constructing the Uniform[0, 1] distribution     15
Extensions of the Extension Theorem     18
Coin tossing and other measures     20
Exercises     23
Section summary     27
Further probabilistic foundations     29
Random variables     29
Independence     31
Continuity of probabilities     33
Limit events     34
Tail fields     36
Exercises     38
Section summary     41
Expected values     43
Simple random variables     43
General non-negative random variables     45
Arbitrary random variables     49
The integration connection     50
Exercises     52
Section summary     55
Inequalities and convergence     57
Various inequalities     57
Convergence of random variables     58
Laws of large numbers     60
Eliminating the moment conditions     61
Exercises     65
Section summary     66
Distributions of random variables     67
Change of variable theorem     67
Examples of distributions     69
Exercises     71
Section summary     72
Stochastic processes and gambling games     73
A first existence theorem     73
Gambling and gambler's ruin     75
Gambling policies     77
Exercises     80
Section summary     81
Discrete Markov chains     83
A Markov chain existence theorem     85
Transience, recurrence, and irreducibility     86
Stationary distributions and convergence     89
Existence of stationary distributions     94
Exercises     98
Section summary     101
More probability theorems     103
Limit theorems     103
Differentiation of expectation     106
Moment generating functions and large deviations     107
Fubini's Theorem and convolution     110
Exercises     113
Section summary     115
Weak convergence     117
Equivalences of weak convergence     117
Connections to other convergence     119
Exercises     121
Section summary     122
Characteristic functions     125
The continuity theorem     126
The Central Limit Theorem     133
Generalisations of the Central Limit Theorem     135
Method of moments     137
Exercises     139
Section summary     142
Decomposition of probability laws     143
Lebesgue and Hahn decompositions     143
Decomposition with general measures     147
Exercises     148
Section summary     149
Conditional probability and expectation     151
Conditioning on a random variable     151
Conditioning on a sub-[sigma]-algebra     155
Conditional variance     157
Exercises     158
Section summary     160
Martingales      161
Stopping times     162
Martingale convergence     168
Maximal inequality     171
Exercises     173
Section summary     176
General stochastic processes     177
Kolmogorov Existence Theorem     177
Markov chains on general state spaces     179
Continuous-time Markov processes     182
Brownian motion as a limit     186
Existence of Brownian motion     188
Diffusions and stochastic integrals     190
Ito's Lemma     193
The Black-Scholes equation     194
Section summary     197
Mathematical Background     199
Sets and functions     199
Countable sets     200
Epsilons and Limits     202
Infimums and supremums     204
Equivalence relations     207
Bibliography     209
Background in real analysis     209
Undergraduate-level probability     209
Graduate-level probability     210
Pure measure theory     210
Stochastic processes     210
Mathematical finance     211
Index     213