Counterexamples in Probability by Stoyanov, Jordan M. -- Read -- Imperial Library of Trantor

Index

Cover About the Author Title Page Copyright Page Dedication Contents Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Basic Notation and Abbreviations Chapter 1. Classes of Random Events and Probabilities

Section 1. Classes of Random Events

1.1. A class of events which is a field but not a σ-field 1.2. A class of events can be closed under finite unions and finite intersections but not under complements 1.3. A class of events which is a semi-field but not a field 1.4. A σ-field of subsets of Ω need not contain all subsets of Ω 1.5. Every σ-field of events is a D-system, but the converse does not always hold 1.6. Sets which are not events in the product σ-field 1.7. The union of a sequence of σ-fields need not be a σ-field

Section 2. Probabilities

2.1. A probability measure which is additive but not σ-additive 2.2. The coincidence of two probability measures on a given class does not always imply their coincidence on the σ-field generated by this class 2.3. On the validity of the Kolmogorov extension theorem in (ℝ∞, ∞) 2.4. There may not exist a regular conditional probability with respect to a given σ-field

Section 3. Independence of Random Events

3.1. Random events with a different kind of dependence 3.2. The pairwise independence of random events does not imply their mutual independence 3.3. The relation P(ABC) = P(A)P(B)P(C) does not always imply the mutual independence of the events A, B, C 3.4. A collection of n + 1 dependent events such that any n of them are mutually independent 3.5. Collections of random events with ‘unusual’ independence/dependence properties 3.6. Is there a relationship between conditional and mutual independence of random events? 3.7. Independence type conditions which do not imply the mutual independence of a set of events 3.8. Mutually independent events can form families which are strongly dependent 3.9. Independent classes of random events can generate σ-fields which are not independent

Section 4. Diverse Properties of Random Events And Their Probabilities

4.1. Probability spaces without non-trivial independent events: totally dependent spaces 4.2. On the Borel-Cantelli lemma and its corollaries 4.3. When can a set of events be both exhaustive and independent? 4.4. How are independence and exchangeability related? 4.5. A sequence of random events which is stable but not mixing

Chapter 2. Random Variables and Basic Characteristics

Section 5. Distribution Functions of Random Variables

5.1. Equivalent random variables are identically distributed but the converse is not true 5.2. If X, Y, Z are random variables on the same probability space, then does not always imply that 5.3. Different distributions can be transformed by different functions to the same distribution 5.4. A function which is a metric on the space of distributions but not on the space of random variables 5.5. On the n-dimensional distribution functions 5.6. The continuity property of one-dimensional distributions may fail in the multi-dimensional case 5.7. On the absolute continuity of the distribution of a random vector and of its components 5.8. There are infinitely many multi-dimensional probability distributions with given marginals 5.9. The continuity of a two-dimensional probability density does not imply that the marginal densities are continuous 5.10. The convolution of a unimodal probability density function with itself is not always unimodal 5.11. The convolution of unimodal discrete distributions is not always unimodal 5.12. Strong unimodality is a stronger property than the usual unimodality 5.13. Every unimodal distribution has a unimodal concentration function, but the converse does not hold

Section 6. Expectations and Conditional Expectations

6.1. On the linearity property of expectations 6.2. An integrable sequence of non-negative random variables need not have a bounded supremum 6.3. A necessary condition which is not sufficient for the existence of the first moment 6.4. A condition which is sufficient but not necessary for the existence of moment of order (–1) of a random variable 6.5. An absolutely continuous distribution need not be symmetric even though all its central odd-order moments vanish 6.6. A property of the moments of random variables which does not have an analogue for random vectors 6.7. On the validity of the Fubini theorem 6.8. A non-uniformly integrable family of random variables 6.9. On the relation E[E(X|Y)] = EX 6.10. Is it possible to extend one of the properties of the conditional expectation? 6.11. The mean-median-mode in equality may fail to hold 6.12. Not all properties of conditional expectations have analogues for conditional medians

Section 7. Independence of Random Variables

7.1. Discrete random variables which are pairwise but not mutually independent 7.2. Absolutely continuous random variables which are pairwise but not mutually independent 7.3. A set of dependent random variables such that any of its subsets consists of mutually independent variables 7.4. Collection of n dependent random variables which are k-wise independent 7.5. An independence-type property for random variables 7.6. Dependent random variables X and Y such that X2 and Y2 are independent 7.7. The independence of random variables in terms of characteristic functions 7.8. The independence of random variables in terms of generating functions 7.9. The distribution of a sum can be expressed by the convolution even if the variables are dependent 7.10. Discrete random variables which are uncorrelated but not independent 7.11. Absolutely continuous random variables which are uncorrelated but not independent 7.12. Independent random variables have zero correlation ratio, but the converse is not true 7.13. The relation E[Y|X] = EY almost surely does not imply that the random variables X and Y are independent 7.14. There is no relationship between the notions of independence and conditional independence 7.15. Mutual independence implies the exchangeability of any set of random variables, but not conversely 7.16. Different kinds of monotone dependence between random variables

Section 8. Characteristic and Generating Functions

Section 9. Infinitely Divisible and Stable Distributions

9.1. A non-vanishing characteristic function which is not infinitely divisible 9.2. If |ø| is an infinitely divisible characteristic function, this does not always imply that ø is also infinitely divisible 9.3. The product of two independent non-negative and infinitely divisible random variables is not always infinitely divisible 9.4. Infinitely divisible products of non-infinitely divisible random variables 9.5. Every distribution without indecomposable components is infinitely divisible, but the converse is not true 9.6. A non-infinitely divisible random vector with infinitely divisible subsets of its coordinates 9.7. A non-infinitely divisible random vector with infinitely divisible linear combinations of its components 9.8. Distributions which are infinitely divisible but not stable 9.9. A stable distribution which can be decomposed into two infinitely divisible but not stable distributions

Section 10. Normal Distribution

10.1. Non-normal bivariate distributions with normal marginals 10.2. If(X1, X2) has a bivariate normal distribution then X1, X2 and X1 + X2 are normally distributed, but not conversely 10.3. A non-normally distributed random vector such that any proper subset of its components consists of jointly normally distributed and mutually independent random variables 10.4. The relationship between two notions: normality and uncorrelatedness 10.5. It is possible that X, Y, X + Y, X – Y are each normally distributed, X and Y are uncorrelated, but (X, Y) is not bivariate normal 10.6. If the distribution of (X1,..., Xn) is normal, then any linear combination and any subset of X1,...,Xn is normally distributed, but there is a converse statement which is not true 10.7. The condition characterizing the normal distribution by normality of linear combinations cannot be weakened 96 10.8. Non-normal distributions such that all or some of the conditional distributions are normal 10.9. Two random vectors with the same normal distribution can be obtained in different ways from independent standard normal random variables 10.10. A property of a Gaussian system may hold even for discrete random variables

Section 11. THE MOMENT PROBLEM

11.1. The moment problem for powers of the normal distribution 11.2. The log normal distribution and the moment problem 11.3. The moment problem for powers of an exponential distribution 11.4. A class of hyper-exponential distributions with an indeterminate moment problem 11.5. Different distributions with equal absolute values of the characteristic functions and the same moments of all orders 11.6. Another class of absolutely continuous distributions which are not determined uniquely by their moments 11.7. Two different discrete distributions on a subset of natural numbers both having the same moments of all orders 11.8. Another family of discrete distributions with the same moments of all orders 11.9. On the relationship between two sufficient conditions for the determination of the moment problem 11.10. The Carleman condition is sufficient but not necessary for the determination of the moment problem 11.11. The Krein condition is sufficient but not necessary for the moment problem to be indeterminate 11.12. An indeterminate moment problem and non-symmetric distributions whose odd-order moments all vanish 11.13. A non-symmetric distribution with vanishing odd-order moments can coincide with the normal distribution only partially

Section 12. Characterization Properties of Some Probability Distributions

12.1. A binomial sum of non-binomial random variables 12.2. A property of the geometric distribution which is not its characterization property 12.3. If the random variables X, Y and their sum X + Y each have a Poisson distribution, this does not imply that X and Y are independent 12.4. The Raikov theorem does not hold without the independence condition 12.5. The Raikov theorem does not hold for a generalized Poisson distribution of order k, k ≥ 2 12.6. A case when the Cramer theorem is not applicable 12.7. A pair of unfair dice may behave like a pair of fair dice 12.8. On two properties of the normal distribution which are not characterizing properties 12.9. Another interesting property which does not characterize the normal distribution 12.10. Can we weaken some conditions under which two distribution functions coincide? 12.11. Does the renewal equation determine uniquely the probability density? 12.12. A property not characterizing the Cauchy distribution 12.13. A property not characterizing the gamma distribution 12.14. An interesting property which does not characterize uniquely the inverse Gaussian distribution

Section 13. Diverse Properties of Random Variables

13.1. On the symmetry property of the sum or the diffference of two symmetric random variables 13.2. When is a mixture of normal distributions infinitely divisible? 13.3. A distribution function can belong to the class IFRA but not to IFR 13.4. A continuous distribution function of the class NBU which is not of the class IFR 13.5. Exchangeable and tail events related to sequences of random variables 13.6. The de Finetti theorem for an infinite sequence of exchangeable random variables does not always hold for a finite number of such variables 13.7. Can we always extend a finite set of exchangeable random variables? 13.8. Collections of random variables which are or are not independent and are or are not exchangeable 13.9. Integrable randomly stopped sums with non-integrable stopping times

Chapter 3. Limit Theorems

Section 14. Various Kinds Of Convergence Of Sequences Of Random Variables

14.1. Convergence and divergence of sequences of distribution functions 14.2. Convergence in distribution does not imply convergence in probability 14.3. Sequences of random variables converging in probability but not almost surely 14.4. On the Borel-Cantelli lemma and almost sure convergence 14.5. On the convergence of sequences of random variables in Lr- sense for different values of r 14.6. Sequences of random variables converging in probability but not in Lr-sense 14.7. Convergence in Lr-sense does not imply almost sure convergence 14.8. Almost sure convergence does not necessarily imply convergence in h^-sense 14.9. Weak convergence of the distribution functions does not imply convergence of the densities 14.10. The convergence and does not always imply that 14.11. The convergence in probability does not always imply that for any function g 14.12. Convergence in variation implies convergence in distribution but the converse is not always true 14.13. There is no metric corresponding to almost sure convergence 14.14. Complete convergence of sequences of random variables is stronger than almost sure convergence 14.15. The almost sure uniform convergence of a random sequence implies its complete convergence, but the converseis not true 14.16. Converging sequences of random variables such that the sequences of the expectations do not converge 14.17. Weak L1-convergence of random variables is weaker than both weak convergence and convergence in L1-sense 14.18. A converging sequence of random variables whose Cesaro means do not converge

Section 15. Laws of Large Numbers

Section 16. Weak Convergence of Probability Measures and Distributions

16.1. Defining classes and classes defining convergence 16.2. In the case of convergence in distribution, do the corresponding probability measures converge for all Borel sets? 16.3. Weak convergence of probability measures need not be uniform 16.4. Two cases when the continuity theorem is not valid 16.5. Weak convergence and Lévy metric 16.6. A sequence of probability density functions can converge in the mean of order 1 without being converging everywhere 16.7. A version of the continuity theorem for distribution functions which does not hold for some densities 16.8. Weak convergence of distribution functions does not imply convergence of the moments 16.9. Weak convergence of a sequence of distributions does not always imply the convergence of the moment generating functions 16.10. Weak convergence of a sequence of distribution functions does not always imply their convergence in the mean

Section 17. Central Limit Theorem

17.1. Sequences of random variables which do not satisfy the central limit theorem 17.2. How is the central limit theorem connected with the Feller condition and the uniform negligibility condition? 17.3. Two ‘equivalent’ sequences of random variables such that one of them obeys the central limit theorem while the other does not 17.4. If the sequence of random variables {Xn} satisfies the central limit theorem, what can we say about the variance of 17.5. Not every interval can be a domain of normal convergence 17.6. The central limit theorem does not always hold for random sums of random variables 17.7. Sequences of random variables which satisfy the integral but not the local central limit theorem

Section 18. Divers Elimit Theorems

18.1. On the conditions in the Kolmogorov three-series theorem 18.2. The independency condition is essential in the Kolmogorov three-series theorem 18.3. The interchange of expectations and infinite summation is not always possible 18.4. A relationship between a convergence of random sequences and convergence of conditional expectations 18.5. The convergence of a sequence of random variables does not imply that the corresponding conditional medians converge 18.6. A sequence of conditional expectations can converge only on a set of measure zero 18.7. When is a sequence of conditional expectations convergent almost surely? 18.8. The Weierstrass theorem for the unconditional convergence of a numerical series does not hold for a series of random variables 18.9. A condition which is sufficient but not necessary for the convergence of a random power series 18.10. A random power series without a radius of convergence in probability 18.11. Two sequences of random variables can obey the same strong law of large numbers but one of them may not be in the domain of attraction of the other 18.12. Does a sequence of random variables always imitate normal behaviour? 18.13. On the Chover law of iterated logarithm 18.14. On record values and maxima of a sequence of random variables

Chapter 4. Stochastic Processes

Section 19. Basic Notions on Stochastic Processes

19.1. Is it possible to find a probability space on which any stochastic process can be defined? 19.2. What is the role of the family of finite-dimensional distributions in constructing a stochastic process with specific properties? 19.3. Stochastic processes whose modifications possess quite different properties 19.4. On the separability property of stochastic processes 19.5. Measurable and progressively measurable stochastic processes 19.6. On the stochastic continuity and the weak L1-continuity of stochastic processes 19.7. Processes which are stochastically continuous but not continuous almostsurely 19.8. Almost sure continuity of stochastic processes and the Kolmogorov condition 19.9. Does the Riemann or Lebesgue integrability of the co variance function ensure the existence of the integral of a stochastic process? 19.10. The continuity of a stochastic process does not imply the continuity of its own generated filtration, and vice versa

Section 20. Markov Processes

20.1. Non-Markov random sequences whose transition functions satisfy the Chapman-Kolmogorov equation 20.2. Non-Markov processes which are functions of Markov processes 20.3. Comparison of three kinds of ergodicity of Markov chains 20.4. Convergence of functions of an ergodic Markov chain 20.5. A useful property of independent random variables which cannot be extended to stationary Markov chains 20.6. The partial coincidence of two continuous-time Markov chains does not imply that the chains are equivalent 20.7. Markov processes, Feller processes, strong Feller processes and relationships between them 20.8. Markov but not strong Markov processes 20.9. Can a differential operator of order k > 2 be an infinitesimal operator of a Markov process?

Section 21. Stationary Processes and Some Related Topics

21.1. On the weak and the strict stationary properties of stochastic processes 21.2. On the strict stationarity of a given order 21.3. The strong mixing property can fail if we consider a functional of a strictly stationary strong mixing process 21.4. A strictly stationary process can be regular but not absolutely regular 21.5. Weak and strong ergodicity of stationary processes 21.6. A measure-preserving transformation which is ergodic but not mixing 21.7. On the convergence of sums of φ-mixing random variables 21.8. The central limit theorem for stationary random sequences

Section 22. Discrete-Time Martingales

22.1. Martingales which are L1-bounded but not L1-dominated 22.2. A property of a martingale which is not preserved under random stopping 22.3. Martingales for which the Doob optional theorem fails to hold 22.4. Every quasimartingale is an amart, but not conversely 22.5. Amarts, martingales in the limit, eventual martingales and relationships between them 22.6. Relationships between amarts, progressive martingales and quasimartingales 22.7. An eventual martingale need not be a game fairer with time 22.8. Not every martingale-like sequence admits a Riesz decomposition 22.9. On the validity of two inequalities for martingales 22.10. On the convergence of submartingales almost surely and in Lr-sense 22.11. A martingale may converge in probability but not almost surely 22.12. Zero-mean martingales which are divergent with a given probability 22.13. More on the convergence of martingales 22.14. A uniformly integrable martingale with a nonintegrable quadratic variation

Section 23. Continuous-Time Martingales

23.1. Martingales which are not locally square integrable 23.2. Every martingale is a weak martingale but the converse is not always true 23.3. The local martingale property is not always preserved under change of time 23.4. A uniformly integrable super martingale which does not belong to class(D) 23.5. YP-bounded local martingale which is not a true martingale 23.6. A sufficient but not necessary condition for a process to be a local martingale 23.7. A square integrable martingale with a non-random characteristic need not be a process with independent increments 23.8. The time-reversal of a semimartingale can fail to be a semimartingale 23.9. Functions of semimartingales which are not semimartingales 23.10. Gaussian processes which are not semimartingales 23.11. On the possibility of representing a martingale as a stochastic integral with respect to another martingale

Section 24. Poisson Process and Wiener Process

24.1. On some elementary properties of the Poisson process and the Wiener process 24.2. Can the Poisson process be characterized by only one of its properties? 24.3. The conditions under which a process is a Poisson process cannot be weakened 24.4. Two dependent Poisson processes whose sum is still a Poisson process 24.5. Multidimensional Gaussian processes which are close to the Wiener process 24.6. On the Wald identities for the Wiener process 24.7. Wald identity and a non-uniformly integrable martingale based on the Wiener process 24.8. On some properties of the variation of the Wiener process 24.9. A Wiener process with respect to different filtrations 24.10. How to enlarge the filtration and preserve the Markov property of the Brownian bridge

Section 25. Diverse Properties of Stochastic Processes

25.1. How can we find the probabilistic characteristics of a function of a stationary Gaussian process? 25.2. Cramer representation, multiplicity and spectral type of stochastic processes 25.3. Weak and strong solutions of stochastic differential equations 25.4. A stochastic differential equation which does not have a strong solution but for which a weak solution exists and is unique

Supplementary Remarks References Appendix

Key Words New References

Index

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