THE THEORY OF PROBABILITY
CONTENTS MEASURE THEORY AND PROBABILITY SOME TRAPS CONCERNING THE UNION OF ?-?ELDS SETS WHICH DO NOT BELONG IN A STRONG SENSE, TO A ?-?ELD SOME CRITERIA FOR UNIFORM INTEGRABILITY WHEN DOES WEAK CONVERGENCE IMPLY THE CONVERGENCE OF EXPECTATIONS? CONDITIONAL EXPECTATION AND THE MONOTONE CLASS THEOREM LP-CONVERGENCE OF CONDITIONAL EXPECTATIONS MEASURE PRESERVING TRANSFORMATIONS ERGODIC TRANSFORMATIONS INVARIANT ?-?ELDS EXTREMAL SOLUTIONS OF (GENERAL) MOMENTS PROBLEMS THE LOG NORMAL DISTRIBUTION IS MOMENTS INDETERMINATE CONDITIONAL EXPECTATIONS AND EQUALITY IN LAW SIMPLI?ABLE RANDOM VARIABLES MELLIN TRANSFORM AND SIMPLI?CATION THERE EXISTS NO FRACTIONAL COVERING OF THE REAL LINE SOLUTIONS FOR CHAPTER 1 INDEPENDENCE AND CONDITIONING INDEPENDENCE DOES NOT IMPLY MEASURABILITY WITH RESPECT TO AN INDEPENDENT COMPLEMENT COMPLEMENT TO EXERCISE 2 1: FURTHER STATEMENTS OF INDEPENDENCE VERSUS MEASURABILITY INDEPENDENCE AND MUTUAL ABSOLUTE CONTINUITY SIZE-BIASED SAMPLING AND CONDITIONAL LAWS THINK TWICE BEFORE EXCHANGING THE ORDER OF TAKING THE SUPREMUM AND INTERSECTION OF ?-?ELDS! EXCHANGEABILITY AND CONDITIONAL INDEPENDENCE: DE FINETTI’S THEOREM ON EXCHANGEABLE ?-?ELDS TOO MUCH INDEPENDENCE IMPLIES CONSTANCY A DOUBLE PARADOXICAL INEQUALITY EULER’S FORMULA FOR PRIMES AND PROBABILITY THE PROBABILITY, FOR INTEGERS, OF BEING RELATIVELY PRIME COMPLETELY INDEPENDENT MULTIPLICATIVE SEQUENCES OF U-VALUED RANDOM VARIABLES BERNOULLI RANDOM WALKS CONSIDERED AT SOME STOPPING TIME COSH, SINH, THE FOURIER TRANSFORM AND CONDITIONAL INDEPENDENCE COSH, SINH, AND THE LAPLACE TRANSFORM CONDITIONING AND CHANGES OF PROBABILITIES RADON–NIKODYM DENSITY AND THE ACCEPTANCE–REJECTION METHOD OF VON NEUMANN NEGLIGIBLE SETS AND CONDITIONING GAMMA LAWS AND CONDITIONING RANDOM VARIABLES WITH INDEPENDENT FRACTIONAL AND INTEGER PARTS TWO CHARACTERIZATIONS OF THE SIMPLE RANDOM WALK SOLUTIONS FOR CHAPTER 2 GAUSSIAN VARIABLES CONSTRUCTING GAUSSIAN VARIABLES FROM, BUT NOT BELONGING TO, A GAUSSIAN SPACE A COMPLEMENT TO EXERCISE 3 1 GAUSSIAN VECTORS AND ORTHOGONAL PROJECTIONS ON THE NEGATIVE MOMENTS OF NORMS OF GAUSSIAN VECTORS QUADRATIC FUNCTIONALS OF GAUSSIAN VECTORS AND CONTINUED FRACTIONS ORTHOGONAL BUT NON-INDEPENDENT GAUSSIAN VARIABLES ISOTROPY PROPERTY OF MULTIDIMENSIONAL GAUSSIAN LAWS THE GAUSSIAN DISTRIBUTION AND MATRIX TRANSPOSITION A LAW WHOSE N-SAMPLES ARE PRESERVED BY EVERY ORTHOGONAL TRANSFORMATION IS GAUSSIAN NON-CANONICAL REPRESENTATION OF GAUSSIAN RANDOM WALKS CONCENTRATION INEQUALITY FOR GAUSSIAN VECTORS DETERMINING A JOINTLY GAUSSIAN DISTRIBUTION FROM ITS CONDITIONAL MARGINALS GAUSSIAN INTEGRATION BY PARTS CORRELATION POLYNOMIALS SOLUTIONS FOR CHAPTER 3 DISTRIBUTIONAL COMPUTATIONS HERMITE POLYNOMIALS AND GAUSSIAN VARIABLES THE BETA–GAMMA ALGEBRA AND POINCAR´E’S LEMMA AN IDENTITY IN LAW BETWEEN RECIPROCALS OF GAMMA VARIABLES THE GAMMA PROCESS AND ITS ASSOCIATED DIRICHLET PROCESSES GAMMA VARIABLES AND GAUSS MULTIPLICATION FORMULAE THE BETA–GAMMA ALGEBRA AND CONVERGENCE IN LAW BETA–GAMMA VARIABLES AND CHANGES OF PROBABILITY MEASURES EXPONENTIAL VARIABLES AND POWERS OF GAUSSIAN VARIABLES MIXTURES OF EXPONENTIAL DISTRIBUTIONS SOME COMPUTATIONS RELATED TO THE LACK OF MEMORY PROPERTY OF THE EXPONENTIAL LAW SOME IDENTITIES IN LAW BETWEEN GAUSSIAN AND EXPONENTIAL VARIABLES SOME FUNCTIONS WHICH PRESERVE THE CAUCHY LAW UNIFORM LAWS ON THE CIRCLE FRACTIONAL PARTS OF RANDOM VARIABLES AND THE UNIFORM LAW NON-IN?NITE DIVISIBILITY AND SIGNED L´EVY–KHINTCHINE REPRESENTATION TRIGONOMETRIC FORMULAE AND PROBABILITY A MULTIDIMENSIONAL VERSION OF THE CAUCHY DISTRIBUTION SOME PROPERTIES OF THE GAUSS TRANSFORM UNILATERAL STABLE DISTRIBUTIONS (1) UNILATERAL STABLE DISTRIBUTIONS (2) UNILATERAL STABLE DISTRIBUTIONS (3) A PROBABILISTIC TRANSLATION OF SELBERG’S INTEGRAL FORMULAE MELLIN AND STIELTJES TRANSFORMS OF STABLE VARIABLES SOLVING CERTAIN MOMENT PROBLEMS VIA SIMPLI?CATION SOLUTIONS FOR CHAPTER 4 CONVERGENCE OF RANDOM VARIABLES AROUND SCHE?´E’S LEMMA CONVERGENCE OF SUM OF SQUARES OF INDEPENDENT GAUSSIAN VARIABLES CONVERGENCE OF MOMENTS AND CONVERGENCE IN LAW BOREL TEST FUNCTIONS AND CONVERGENCE IN LAW CONVERGENCE IN LAW OF THE NORMALIZED MAXIMUM OF CAUCHY VARIABLES LARGE DEVIATIONS FOR THE MAXIMUM OF GAUSSIAN VECTORS A LOGARITHMIC NORMALIZATION A ?N LOG N NORMALIZATION THE CENTRAL LIMIT THEOREM INVOLVES CONVERGENCE IN LAW, NOT IN PROBABILITY CHANGES OF PROBABILITIES AND THE CENTRAL LIMIT THEOREM CONVERGENCE IN LAW OF STABLE(?) VARIABLES, AS ? ?0 FINITE-DIMENSIONAL CONVERGENCE IN LAW TOWARDS BROWNIAN MOTION THE EMPIRICAL PROCESS AND THE BROWNIAN BRIDGE THE FUNCTIONAL LAW OF LARGE NUMBERS THE POISSON PROCESS AND THE BROWNIAN MOTION BROWNIAN BRIDGES CONVERGING IN LAW TO BROWNIAN MOTIONS AN ALMOST SURE CONVERGENCE RESULT FOR SUMS OF STABLE RANDOM VARIABLES SOLUTIONS FOR CHAPTER 5 RANDOM PROCESSES JEULIN’S LEMMA DEALS WITH THE ABSOLUTE CONVERGENCE OF INTEGRALS OF RANDOM PROCESSES FUNCTIONS OF BROWNIAN MOTION AS SOLUTIONS TO SDES; THE EXAMPLE OF ?(X) = SINH(X) BOUGEROL’S IDENTITY AND SOME BESSEL VARIANTS DOL´EANS–DADE EXPONENTIALS AND THE MARUYAMA–GIRSANOV–VAN SCHUPPEN–WONG THEOREM REVISITED THE RANGE PROCESS OF BROWNIAN MOTION SYMMETRIC L´EVY PROCESSES RE?ECTED AT THEIR MINIMUM AND MAXIMUM; E CS´AKI’S FORMULAE FOR THE RATIO OF BROWNIAN EXTREMES IN?NITE DIVISIBILITY WITH RESPECT TO TIME A TOY EXAMPLE FOR WESTWATER’S RENORMALIZATION SOME ASYMPTOTIC LAWS OF PLANAR BROWNIAN MOTION WINDINGS OF THE THREE-DIMENSIONAL BROWNIAN MOTION AROUND A LINE CYCLIC EXCHANGEABILITY PROPERTY AND UNIFORM LAW RELATED TO THE BROWNIAN BRIDGE LOCAL TIME AND HITTING TIME DISTRIBUTIONS FOR THE BROWNIAN BRIDGE PARTIAL ABSOLUTE CONTINUITY OF THE BROWNIAN BRIDGE DISTRIBUTION WITH RESPECT TO THE BROWNIAN DISTRIBUTION A BROWNIAN INTERPRETATION OF THE DUPLICATION FORMULA FOR THE GAMMA FUNCTION SOME DETERMINISTIC TIME-CHANGES OF BROWNIAN MOTION A NEW PATH CONSTRUCTION OF BROWNIAN AND BESSEL BRIDGES RANDOM SCALING OF THE BROWNIAN BRIDGE TIME-INVERSION AND QUADRATIC FUNCTIONALS OF BROWNIAN MOTION; L´EVY’S STOCHASTIC AREA FORMULA QUADRATIC VARIATION AND LOCAL TIME OF SEMIMARTINGALES GEOMETRIC BROWNIAN MOTION 0-SELF SIMILAR PROCESSES AND CONDITIONAL EXPECTATION A TAYLOR FORMULA FOR SEMIMARTINGALES; MARKOV MARTINGALES AND ITERATED IN?NITESIMAL GENERATORS A REMARK OF D WILLIAMS: THE OPTIONAL STOPPING THEOREM MAY HOLD FOR CERTAIN “NON-STOPPING TIMES” STOCHASTIC A?NE PROCESSES, ALSO KNOWN AS “HARNESSES” MORE ON HARNESSES A MARTINGALE “IN THE MEAN OVER TIME” IS A MARTINGALE A REINFORCEMENT OF EXERCISE 6 26 SOME PAST-AND-FUTURE BROWNIAN MARTINGALES ADDITIVE AND MULTIPLICATIVE MARTINGALE DECOMPOSITIONS OF BROWNIAN MOTION SOLUTIONS FOR CHAPTER 6 WHERE IS THE NOTION N DISCUSSED? FINAL SUGGESTIONS: HOW TO GO FURTHER? REFERENCES