Counterexamples in Probability

CONTENTS ......Page 1
Preface to the Second Edition ......Page 27
Perface to the First Edition ......Page 29
Part 1 Classes of Random Events and Probabilities ......Page 32
1.1 A class of events which is a field but not a sigma field  ......Page 34
1.4 A sigma field of subsets of S need not contain all subsets of S ......Page 35
1.7 The union of a sequence of sigma fields need not be sigma field ......Page 36
2 Probabilities ......Page 37
2.1 A probability measure which is additive but not sigma additive ......Page 38
2.3 On the validity of the Kolmogorov extension theorem in (R,B) ......Page 39
2.4 There may not exist a regular conditional probability with respect to a given sigma field ......Page 41
3 Independence of Random Events ......Page 42
3.1 Random events with a different kind of dependence ......Page 43
3.2 The pairwise independence of random events does not imply their mutual independence ......Page 44
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 ......Page 45
3.4 A collection of n+1 dependent events such that any n of them are mutually independent ......Page 46
3.5 Collections of random events with \'ususual\' independence/dependence properties ......Page 47
3.6 Is there a relationship between conditional and mutual independece of random events ......Page 49
3.8 Mutually independent events can form families which are strongly dependent ......Page 50
3.9 Independence classes of random events can generate sigma fields which are not independent ......Page 51
4.1 Probability spaces without non-trivial independent events: totally dependent spaces ......Page 52
4.2 On the Borel-Cantelli lemma and its corollaries ......Page 53
4.3 When can a set of events be both exhaustive and independent ......Page 54
4.4 How are independence and exchangeability related? ......Page 55
4.5 A sequence of random events which is stable but not mixing ......Page 56
Part 2 Random Variables and Basic Characteristics ......Page 58
5 Distributions of Functions of Random Variables ......Page 60
5.2 If X,Y,Z are random variables on the same probabiity space then X=Y does not imply XZ=YZ ......Page 62
5.3 Different distributions can be transformed by different functions to the same distribution ......Page 63
5.5 On the n-dimensional distribution functions ......Page 64
5.6 The continuity property of one-dimensional distributions may fail in the multi-dimensional case ......Page 65
5.7 On the absolute continuity of the distribution of a random vector and of its components ......Page 66
5.8 There are infinitely many multi-dimensional probability distributions with given marginals ......Page 67
5.9 The continutiy of a two-dimensional probability density does not imply that the marginal densitites are continuous ......Page 68
5.10 The convolution of a unimodal probability density function with itself is not always unimodal ......Page 69
5.12 Strong unimodality is a stronger property than the usual unimodality ......Page 71
5.13 Every unimodal distribution has a unimodal concentration function, but the converse does not hold ......Page 72
6 Expectations and Conditonal Expectations ......Page 73
6.1 On the linearity property of expectations ......Page 75
6.3 A necessary condition which is not sufficient for the existence of the first moment ......Page 76
6.4 A condition which is sufficient but not necessary for the existence of the moment of order (-1) of a random variable ......Page 77
6.6 A property of the moments of random variables which does not have an analogue for random vectors ......Page 78
6.8 A non-uniformly integrable family of random variables ......Page 79
6.10 Is it possible to extend one of the properties of the conditional expectations? ......Page 80
6.11 The mean-median-mode inequality may fail to hold ......Page 81
7 Independence of Random Variables ......Page 82
7.2 Absolutely continuous random variables which are pairwise but not mutually independent ......Page 84
7.3 A set of dependent random variables such that any of its subsets consists of mutually independent variables ......Page 85
7.4 Collection of n dependent random variables which are m-wise independent ......Page 87
7.6 Dependent random variables X and Y such that X^2 and Y^2 are independent ......Page 89
7.7 The independence of random variables in terms of characteristic functions ......Page 91
7.8 The independence of random variables in terms of generating functions ......Page 92
7.9 The distribution of a sum can be expressed by the convolution even if the variables are dependent ......Page 93
7.10 Discrete random variables which are uncorrelated but not independent ......Page 94
7.12 Independent random variables have zero correlation ratio, but the converse is not true ......Page 95
7.14 There is no relationship between the notions of independence and conditional independence ......Page 96
7.16 Different kinds of monotone dependence between random variables ......Page 98
8 Characteristic and Generating Functions ......Page 99
8.1 Different characteristic functions which coincide on a finite interval but not on the whole real line ......Page 101
8.2 Discrete and absolutely continuous distributions can have characteristic functions coinciding on the interval [-1,1] ......Page 102
8.4 The ratio of two characteristic functions need not be a characteristic function ......Page 103
8.5 The factorization of a characteristic function into decomposable factors may not be unique ......Page 104
8.6 An absolutely continuous distribution can have a characteristic function whcih is not absolutely integrable ......Page 105
8.8 An absolutely continuous distribution wihtout expectation ......Page 106
8.9 The convolution of two indecomposable distributions can even have a normal component  ......Page 107
8.10 Does the existence of all moments of a distribution guarantee the analyticity of its characteristic and moment generating functions? ......Page 108
9.1 A non-vanishing characteristic function which is not infinitely divisible ......Page 109
9.2 If |phi| is an infinitely divisible characteristic function, this does not always imply that phi is also infinitely divisible ......Page 111
9.3 The product of two independent non-negative ......Page 112
9.4 Infinitely divisible products of non-infinitely divisible random variables ......Page 113
9.6 A non-infinitely divisible random vector with infinitely divisible subsets of its coordinates ......Page 114
9.7 A non-infinitely divisible random vector with infinitely divisible linear combinations of its components ......Page 115
9.8 Distributions which are infinitely divisible but not stable ......Page 116
9.9 A stable distribution which can be decomposed into two infinitely divisible but not stable distributions ......Page 117
10 Normal Distribution ......Page 118
10.1 Non-normal bivariate distributions with normal marginals ......Page 119
10.2 If (X1,X2) has a bivariate normal distribution then X1,X2 and X1+X2 are normally distributed, but not conversely ......Page 120
10.4 The relationshiop between two notions: normality and uncorrelatedness ......Page 122
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 ......Page 125
10.8 Non-normal distributions such that all or some of the conditional distributions are normal ......Page 127
10.9 Two random vectors with the same normal distribution ......Page 129
10.10 A property of a Gaussian system may hold even for discrete random variables ......Page 130
11 The Moment Problem ......Page 131
11.1 The moment problem for powers of the normal distribution ......Page 132
11.2 The lognormal distribution and the moment problem ......Page 133
11.4 A class of hyper-exponential distributions with an indeterminate moment problem ......Page 136
11.5 Different distributions with equal absolute values of the characteristic functions and the same moments of all orders ......Page 138
11.6 Another class of absolutely continuous distributions which are not determined uniquely by their moments ......Page 139
11.7 Two different discrete distributions on a subset of natural numbers both having the same moments of all orders ......Page 140
11.9 On the relationship between two sufficient conditions for the determination of the moment problem ......Page 141
11.10 The Carleman condition is sufficient but not necessary for the determination of the moment problem ......Page 144
11.11 The Krein condition is sufficent but not necessary for the moment problem to be indeterminate ......Page 145
11.13 A non-symmetric distribution with vanishing odd-order moments can coincide with the normal distribution only partially ......Page 146
12 Charaterization Properties of Probability Distributions ......Page 147
12.1 A binomial sum of non-binomial random variables ......Page 148
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 ......Page 149
12.4 The Raikov theorem does not hold without the independence condition ......Page 150
12.5 The Raikov theorem does not hold for a generalized Poisson distribution of order k, K>=2 ......Page 151
12.7 A pair of unfair dice behave like a pair of fair dice ......Page 152
12.8 On two properties of the normal distribution which are not characterizing properties ......Page 153
12.9 Another interesting property which does not characterize the normal distribution ......Page 156
12.10 Can we weaken some conditions under which two distribution functions coincide? ......Page 157
12.11 Does the renewal equation determine uniquely the probability density? ......Page 158
12.13 A property not characterizing the gamma distribution ......Page 159
12.14 An interesting property which does not characterize uniquely the inverse Gaussian distribution ......Page 160
13.1 On the symmetry property of the sum or the difference of two symmetric random variables ......Page 161
13.2 When is a mixture of normal distributions infinitely divisible? ......Page 163
13.3 A distribution functiion can belong to the class of IFRA but not IFR ......Page 164
13.5 Exchangeabiity and tail events related to sequences of random variables ......Page 165
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 ......Page 167
13.7 Can we always extend a finite set of exchangeable random variables ......Page 168
13.9 Integrable randomly stopped sums with non-integrable stopping times ......Page 169
Part 3 Limit Theorems ......Page 171
14 Various Kinds of Convergence of Sequences of Random Variables ......Page 173
14.1 Convergence and divergence of sequences of distribution functions ......Page 174
14.2 Convergence in distribution does not imply convergence in probability ......Page 175
14.3 Sequences of random variables converging in probability but not almost surely ......Page 176
14.4 On the Borel-Cantelli lemma and almost sure convergence ......Page 177
14.6 Sequences of random variables converging in probability but not in the Lr-sense ......Page 178
14.7 Convergence in Lr-sense does not imply almost sure convergence ......Page 179
14.8 Almost sure convergence does not necessarily imply convergence in Lr-sense ......Page 180
14.9 Weak convergence of the distribution functions does not imply convergence of the densities ......Page 181
14.11 The convergence in probability Xn->X does not always imply that g(Xn)->g(X) for any function g ......Page 182
14.12 Convergence in variation imlies convergence in distribution but the converse is not always true ......Page 183
14.14 Complete convergence of sequences of random variables is stronger than almost sure convergence ......Page 185
14.16 Converging sequences of random variables such that the sequences of the expectations do not converge ......Page 186
14.17 Weak L1-convergence of random variables is weaker than both weak convergence and convergence in L1-sense ......Page 187
14.18 A converging sequence of random variables whose Cesaro means do not converge ......Page 188
15 Laws of Large Numbers ......Page 189
15.1 The Markov condition is sufficient but not necessary for the weak law of large numbers ......Page 191
15.2 The Kolmogorov condition for arbitrary random variables is sufficient but not necessary for the strong law of large numbers ......Page 192
15.3 A sequence of independent discrete random variables satisfying the weak but not the strong law of large numbers ......Page 193
15.4 A sequence of independent absolutely continuous random variables satisfying the weak but not the strong law of large numbers ......Page 194
15.6 More on the strong law of large numbers without the Kolmogorov condition ......Page 195
15.7 Two \'near\' sequences of random variables such that the strong law of large numbers holds for one of them and does not hold for the other ......Page 196
15.9 The uniform boundedness of the first moments of a tight sequence of random numbers is not sufficient for the strong law of large numbers ......Page 197
15.10 The arithmetic means of a random sequence can converge in probability even if the strong law of large numbers fails to hold ......Page 198
15.11 The weighted averages of a sequence of random variables can converge even if the law of large numbers does not hold ......Page 199
15.12 The law of large numbers with a special choice of norming constants ......Page 200
16 Weak Convergence of Probability Measures and Distributions ......Page 201
16.1 Defining classes and classes defining convergence ......Page 203
16.3 Weak convergence of probability measures need not be uniform ......Page 204
16.4 Two cases when the continuity theorem is not valid ......Page 206
16.5 Weak convergence and Levy metric ......Page 207
16.6 A sequence of probability density functions can converge in the mean of order 1 wihtout being converging everywhere ......Page 208
16.8 Weak convergence of distribution functions does not imply convergence of the moments ......Page 209
16.10 Weak convergence of a sequence of distribution functions does not always imply their convergence in the mean ......Page 212
17 Central Limit Theorem ......Page 213
17.1 Sequences of random variables which do not satisfy the central limit theorem ......Page 214
17.3 Two \'equivalent\' sequences of random variables such that one of them obeys the central limit theorem while the other does not  ......Page 216
17.4 If the sequence of random variables {Xn} satisfies the central limit theorem, what can we say about the variance of Sn/SQRT(VarSn) ......Page 217
17.5 Not every interval can be a domain of normal convergence ......Page 218
17.7 Sequences of random variables which satisfy the integral but not the local central limit theorem ......Page 219
18.1 On the conditions in the Kolmogorov three-series theorem ......Page 222
18.2 The independency condition is essential in the Kolmogorov three-series theorem ......Page 223
18.4 A relationship between a convergence of random sequences and convergence of conditional expectations ......Page 225
18.5 The convergence of a sequence of random variables does not imply that the corresponding conditional medians converge ......Page 226
18.7 When is a sequence of condtional expectations convergent almost surely? ......Page 227
18.8 The Weierstrass theorem for the unconditional convergence of a numerical series does not hold for a series of random variables ......Page 228
18.9 A condition which is sufficient but not necessary for the convergence of a random power series ......Page 229
18.10 A random power series without a radius of convergence in probability ......Page 230
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 ......Page 231
18.12 Does a sequence of random variables alays imitate normal behavior? ......Page 232
18.13 On the Chover law of iterated logarithm ......Page 234
18.14 On record values and maxima of a sequence of random variables ......Page 235
Part 4 Stochastic Processes ......Page 236
19 Basic Notions on Stochastic Processes ......Page 238
19.1 Is it possible to find a probability space on which any stochastic process can be defined? ......Page 239
19.2 What is the role of the family of finite-dimensional distributions in constructing a stochastic process with specific properties? ......Page 240
19.4 On the separability property of stochastic processes ......Page 241
19.5 Measurable and progressively measurable stochastic processes ......Page 243
19.6 On the stochastic continuity and the weak L1-continuity of stochastic processes ......Page 246
19.8 Almost sure continuity of stochastic processes and the Kolmogorov condition ......Page 248
19.9 Does the Riemann or Lebesgue integrability of the covariance function ensure the existence of the integral of a stochastic process? ......Page 249
19.10 The continuity of a stochastic process does not imply the continuity of its own generated filtration, and vice versa ......Page 252
20 Markov Processes ......Page 253
20.2 Non-Markov processes which are functions of Markov processes ......Page 255
20.3 Comparison of three kinds of ergocity of Markov chains ......Page 256
20.4 Convergence of functions of an ergodic Markov chain ......Page 261
20.5 A useful porperty of independent random variables which cannot be extended to stationary Markov chains ......Page 262
20.7 Markov processes, Feller processes, strong Feller processes and relationships between them ......Page 263
20.8 Markov but not strong Markov processes ......Page 265
20.9 Can a differential operator of order k>2 be an infinitesimal operator of a Markov process ......Page 267
21 Stationary Processes and Some Related Topics ......Page 268
21.1 On the weak and the strict stationarity proporties of stochastic processes ......Page 269
21.2 On the strict stationarity of a given order ......Page 270
21.3 The strong mixing property can fail if we consider a fundamental of a strictly stationary strong mixing process ......Page 271
21.4 A strictly stationary process can be regular but not absolutely regular ......Page 272
21.6 A measure-preserving transformation which is ergodic but not mixing ......Page 273
21.8 The central limit theorem for stationary random processes ......Page 277
22 Discrete-Time Martingales ......Page 279
22.1 Martingales which are L1-bounded but not L1-dominated ......Page 280
22.2 A property of a martingale which is not preserved under random stopping ......Page 281
22.3 Martingales for which the Doob optional theorem fails to hold ......Page 283
22.4 Every quasimartingale is an amart, but not conversely ......Page 284
22.5 Amarts, martingales in the limit, eventual martingales and relationships between them ......Page 285
22.6 Relationships between amarts, progressive martingales and quasimartingales ......Page 286
22.8 Not every martingale-like sequence admits a Riesz decomposition ......Page 287
22.10 On the convergence of submartingales almost surely and in L1-sense ......Page 288
22.11 A martingale may converge in probability but not almost surely ......Page 290
22.13 More on the convergence of martingales ......Page 292
22.14 A uniformly integrable martingale with a nonintegrable quadratic variation ......Page 294
23 Continuous-Time Martingales ......Page 296
23.1 Martingales which are not locally square integrable ......Page 297
23.2 Every martingale is a weak martingale but the converse is not always true ......Page 298
23.3 The local martingale property is not always preserved under change of time ......Page 299
23.4 A uniformly integrable supermartingale which does not belong to class (D) ......Page 300
23.5 Lp-bounded local martingale which is not a true martingale ......Page 301
23.6 A sufficient but not necessary condition for a process to be a local martingale ......Page 303
23.7 A square integrable martingale with a non-random characteristic need not be a process with independent increments ......Page 304
23.9 Functions of semimartingales which are not semimartingales ......Page 305
23.10 Gaussian processes which are not semimartingales ......Page 306
23.11 On the possibility of representing a martingale as a stochastic integral with respect to another martingale ......Page 308
24 Poisson Processes and Wiener Processes ......Page 309
24.1 On some elementary properties of the Poisson process and the Wiener process ......Page 310
24.2 Can the Poisson process be characterized by only one of its properties? ......Page 312
24.3 The conditions under which a process is a Poisson process cannot be weakened ......Page 313
24.4 Two dependent Poisson processes whose sum is still a Poisson process ......Page 315
24.5 Muiltidimensional Gaussian processes which are close to the Wiener process ......Page 316
24.6 On the Wald identities for the Wiener process ......Page 317
24.7 Wald identity and a non-uniformly integrable martingale based on the Wiener process ......Page 319
24.8 On some properties of the variation of the Wiener process ......Page 320
24.9 A Wiener process with respect to different filtrations ......Page 322
24.10 How to enlarge the filtration and preserve the Markov property of the Brownian bridge ......Page 323
25 Diverse Properties of Stochastic Processes ......Page 324
25.1 How can we find the probabilistic charactersitics of a function of a stationary Gaussian process? ......Page 325
25.2 Cramer representation, multiplicity and spectral type of stochastic processes ......Page 326
25.3 Weak and strong solutions of stochastic differential equations ......Page 329
25.4 A stochastic differential equation which does not have a strong solution but for which a weak solution exists and is unique ......Page 331
Supplementary Remarks ......Page 333
References ......Page 345