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Let \( \xi \) and \( \eta \) be independent identically distributed random variables, each taking the values 1 and 0 with probabilities \( p \) and \( q \) . For \( k = 0,1,2 \), let us find the conditional probability \( \mathrm{P}\left( {\xi + \eta = k \mid \eta }\right) \) of the event \( A = \{ \omega : \xi + \eta ... | To do this, we first notice the following useful general fact: if \( \xi \) and \( \eta \) are independent random variables with respective values \( x \) and \( y \), then\n\n\[ \mathrm{P}\left( {\xi + \eta = z \mid \eta = y}\right) = \mathrm{P}\left( {\xi + y = z}\right) . \]\n\nIn fact,\n\n\[ \mathrm{P}\left( {\xi +... | Yes |
If \( \mathcal{D} \) is the trivial decomposition, \( \mathcal{D} = \{ \Omega \} \), then \( \eta \) is \( \mathcal{D} \)-measurable if and only if \( \eta \equiv C \), where \( C \) is a constant. | Suppose that the random variable \( \eta \) is \( \mathcal{D} \)-measurable. Then\n\n\[ \mathrm{E}\left( {{\xi \eta } \mid \mathcal{D}}\right) = \eta \mathrm{E}\left( {\xi \mid \mathcal{D}}\right) \]\n\nand in particular\n\n\[ \mathrm{E}\left( {\eta \mid \mathcal{D}}\right) = \eta \;\left( {\mathrm{E}\left( {\eta \mid ... | No |
Let us find \( \mathrm{E}\left( {\xi + \eta \mid \eta }\right) \) for the random variables \( \xi \) and \( \eta \) considered in Example 1. | By (22) and (23), \[ \mathrm{E}\left( {\xi + \eta \mid \eta }\right) = \mathrm{E}\xi + \eta = p + \eta . \] This result can also be obtained by starting from (8): \[ \mathrm{E}\left( {\xi + \eta \mid \eta }\right) = \mathop{\sum }\limits_{{k = 0}}^{2}k\mathrm{P}\left( {\xi + \eta = k \mid \eta }\right) = p\left( {1 - \... | Yes |
Let \( \xi \) and \( \eta \) be independent and identically distributed random variables. Then\n\n\[ \mathrm{E}\left( {\xi \mid \xi + \eta }\right) = \mathrm{E}\left( {\eta \mid \xi + \eta }\right) = \frac{\xi + \eta }{2}. \] | In fact, if we assume for simplicity that \( \xi \) and \( \eta \) take the values \( 1,2,\ldots, m \), we find \( \left( {1 \leq k \leq m,2 \leq l \leq {2m}}\right) \)\n\n\[ \mathrm{P}\left( {\xi = k \mid \xi + \eta = l}\right) = \frac{\mathrm{P}\left( {\xi = k,\xi + \eta = l}\right) }{\mathrm{P}\left( {\xi + \eta = l... | Yes |
Lemma 1. Let \( a \) and \( b \) be nonnegative integers, \( a - b > 0 \) and \( k = a + b \) . Then\n\n\[ \n{L}_{k}\left( {{S}_{1} > 0,\ldots ,{S}_{k - 1} > 0,{S}_{k} = a - b}\right) = \frac{a - b}{k}{C}_{k}^{a}.\n\] | Proof. In fact,\n\n\[ \n{L}_{k}\left( {{S}_{1} > 0,\ldots ,{S}_{k - 1} > 0,{S}_{k} = a - b}\right) \n\]\n\n\[ \n= {L}_{k}\left( {{S}_{1} = 1,{S}_{2} > 0,\ldots ,{S}_{k - 1} > 0,{S}_{k} = a - b}\right) \n\]\n\n\[ \n= {L}_{k}\left( {{S}_{1} = 1,{S}_{k} = a - b}\right) - {L}_{k}\left( {{S}_{1} = 1,{S}_{k} = a - b}\right. ... | Yes |
Let \( {\eta }_{1},\ldots ,{\eta }_{n} \) be independent Bernoulli random variables with\n\n\[ \mathrm{P}\left( {{\eta }_{k} = 1}\right) = \mathrm{P}\left( {{\eta }_{k} = - 1}\right) = \frac{1}{2}, \]\n\n\[ {S}_{k} = {\eta }_{1} + \cdots + {\eta }_{k}\;\text{ and }\;{\mathcal{D}}_{k} = {\mathcal{D}}_{{\eta }_{1},\ldots... | In fact, \( {S}_{k} \) is \( {\mathcal{D}}_{k} \)-measurable, and by (12) and (18) of Sect. 8\n\n\[ \mathrm{E}\left( {{S}_{k + 1} \mid {\mathcal{D}}_{k}}\right) = \mathrm{E}\left( {{S}_{k} + {\eta }_{k + 1} \mid {\mathcal{D}}_{k}}\right) \]\n\n\[ = \mathrm{E}\left( {{S}_{k} \mid {\mathcal{D}}_{k}}\right) + \mathrm{E}\l... | Yes |
Let \( \eta \) be a random variable, \( {\mathcal{D}}_{1} \preccurlyeq \cdots \preccurlyeq {\mathcal{D}}_{n} \), and \[ {\xi }_{k} = \mathrm{E}\left( {\eta \mid {\mathcal{D}}_{k}}\right) \] Then the sequence \( \xi = {\left( {\xi }_{k},{\mathcal{D}}_{k}\right) }_{1 \leq k \leq n} \) is a martingale. | In fact, it is evident that \( \mathrm{E}\left( {\eta \mid {\mathcal{D}}_{k}}\right) \) is \( {\mathcal{D}}_{k} \) -measurable, and by (20) of Sect. 8 \[ \mathrm{E}\left( {{\xi }_{k + 1} \mid {\mathcal{D}}_{k}}\right) = \mathrm{E}\left\lbrack {\mathrm{E}\left( {\eta \mid {\mathcal{D}}_{k + 1}}\right) \mid {\mathcal{D}}... | Yes |
Let \( {\eta }_{1},\ldots ,{\eta }_{n} \) be a sequence of independent identically distributed random variables, \( {S}_{k} = {\eta }_{1} + \cdots + {\eta }_{k} \), and \( {\mathcal{D}}_{1} = {\mathcal{D}}_{{S}_{n}},{\mathcal{D}}_{2} = {\mathcal{D}}_{{S}_{n},{S}_{n - 1}},\ldots ,{\mathcal{D}}_{n} = \) \( {\mathcal{D}}_... | In the first place, it is clear that \( {\mathcal{D}}_{k} \preccurlyeq {\mathcal{D}}_{k + 1} \) and \( {\xi }_{k} \) is \( {\mathcal{D}}_{k} \) -measurable. Moreover, we have by symmetry, for \( j \leq n - k + 1 \) ,\n\n\[ \n\mathrm{E}\left( {{\eta }_{j} \mid {\mathcal{D}}_{k}}\right) = \mathrm{E}\left( {{\eta }_{1} \m... | Yes |
Example 5. Let \( {\eta }_{1},\ldots ,{\eta }_{n} \) be independent Bernoulli random variables with\n\n\[\n\mathrm{P}\left( {{\eta }_{i} = + 1}\right) = \mathrm{P}\left( {{\eta }_{i} = - 1}\right) = \frac{1}{2},\n\]\n\n\( {S}_{k} = {\eta }_{1} + \cdots + {\eta }_{k} \) . Let \( A \) and \( B \) be integers, \( A < 0 < ... | 3. It follows from the definition of a martingale that the expectation \( \mathrm{E}{\xi }_{k} \) is the same for every \( k \) :\n\n\[\n\mathrm{E}{\xi }_{k} = \mathrm{E}{\xi }_{1}\n\]\n\nIt turns out that this property persists if time \( k \) is replaced by a stopping time. In order to formulate this property we intr... | No |
Let \( X = \{ 0,1,2\} \) and \[ \begin{Vmatrix}{p}_{ij}\end{Vmatrix} = \left( \begin{matrix} 1 & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{2}{3} & 0 & \frac{1}{3} \end{matrix}\right) \] | Here state 0 is said to be absorbing: if the particle gets into this state it remains there, since \( {p}_{00} = 1 \) . From state 1 the particle goes to the adjacent states 0 or 2 with equal probabilities; state 2 has the property that the particle remains there with probability \( \frac{1}{3} \) and goes to state 0 w... | No |
Let \( X = \{ 0, \pm 1,\ldots , \pm N\} ,{p}_{0} = 1,{p}_{NN} = {p}_{-N, - N} = 1 \), and, for \( \left| i\right| < N \)\n\n\[ \n{p}_{ij} = \left\{ \begin{array}{ll} p, & j = i + 1 \\ q, & j = i - 1 \\ 0 & \text{ otherwise } \end{array}\right. \]\n\nThe transitions corresponding to this chain can be presented graphical... | In fact, if \( {\eta }_{1},{\eta }_{2},\ldots ,{\eta }_{n} \) are independent Bernoulli random variables with \( \mathrm{P}\left( {{\eta }_{i} = }\right. \) \( + 1) = p,\mathrm{P}\left( {{\eta }_{i} = - 1}\right) = q,{S}_{0} = 0 \) and \( {S}_{k} = {\eta }_{1} + \cdots + {\eta }_{k} \) the amounts won by the first play... | Yes |
At a taxi stand let taxis arrive at unit intervals of time (one at a time). If no one is waiting at the stand, the taxi leaves immediately. Let \( {\eta }_{k} \) be the number of passengers who arrive at the stand at time \( k \), and suppose that \( {\eta }_{1},\ldots ,{\eta }_{n} \) are independent random variables. ... | In other words,\n\n\[ {\xi }_{k + 1} = {\left( {\xi }_{k} - 1\right) }^{ + } + {\eta }_{k + 1},\;0 \leq k \leq n - 1, \]\n\nwhere \( {a}^{ + } = \max \left( {a,0}\right) \), and therefore the sequence \( \xi = \left( {{\xi }_{0},\ldots ,{\xi }_{n}}\right) \) is a Markov chain. | Yes |
Consider a homogeneous Markov chain with the two states 0 and 1 and the transition matrix \[ \mathbb{P} = \left( \begin{array}{ll} {p}_{00} & {p}_{01} \\ {p}_{10} & {p}_{11} \end{array}\right) \] | It is easy to calculate that \[ {\mathbb{P}}^{2} = \left( \begin{array}{ll} {p}_{00}^{2} + {p}_{01}{p}_{10} & {p}_{01}\left( {{p}_{00} + {p}_{11}}\right) \\ {p}_{10}\left( {{p}_{00} + {p}_{11}}\right) & {p}_{11}^{2} + {p}_{01}{p}_{10} \end{array}\right) \] and (by induction) \[ {\mathbb{P}}^{n} = \frac{1}{2 - {p}_{00} ... | Yes |
Theorem 2. Let \( \xi = \left( {{\xi }_{0},\ldots ,{\xi }_{n}}\right) \) be a homogeneous Markov chain with transition matrix \( \begin{Vmatrix}{p}_{ij}\end{Vmatrix},\tau \) a stopping time (with respect to \( {\mathcal{D}}^{\xi } \) ), \( B \in {\mathcal{B}}_{\tau }^{\xi } \) and \( A = \{ \omega : \tau + l \leq n\} \... | Proof. For the sake of simplicity, we give the proof only for the case \( l = 1 \) . Since \( B \cap \left( {\tau = k}\right) \in {\mathcal{B}}_{k}^{\xi } \), we have, according to (29),\n\n\[ \mathbf{P}\left\{ {{\xi }_{\tau + 1} = {a}_{1}, A \cap B \cap \left( {{\xi }_{\tau } = {a}_{0}}\right) }\right\} \]\n\n\[ = \ma... | Yes |
Consider a group of 30 students \( \left( {N\left( \Omega \right) = {30}}\right) \) . In this group 10 students study the foreign language \( A\left( {N\left( A\right) = {10}}\right) \) and 15 students study the language \( B \) \( \left( {N\left( B\right) = {15}}\right) \), while 5 of them study both languages \( \lef... | Clearly, this number is \( N\left( {\bar{A}\bar{B}}\right) \) . According to (6)\n\n\[ N\left( {\bar{A}\bar{B}}\right) = {30} - \left\lbrack {{10} + {15} - 5}\right\rbrack = {10}. \]\n\nThus there are 10 students who do not study either of these languages. | Yes |
How many integers between 1 and 300 (A) are not divisible by 3 ? | (A) Let \( N\left( A\right) \) be the number of integers (in the interval \( \left\lbrack {1,\ldots ,{300}}\right\rbrack \) ) divisible by 3. Clearly, \( N\left( A\right) = \frac{1}{3} \cdot {300} = {100} \) . Therefore the number of integers which are not divisible by 3 is \( N\left( \bar{A}\right) = N\left( \Omega \r... | Yes |
Lemma 1. Let \( \mathcal{E} \) be a collection of subsets of \( \Omega \) . Then there are the smallest algebra \( \alpha \left( \mathcal{E}\right) \) and the smallest \( \sigma \) -algebra \( \sigma \left( \mathcal{E}\right) \) containing all the sets that are in \( \mathcal{E} \) . | Proof. The class \( {\mathcal{F}}^{ * } \) of all subsets of \( \Omega \) is a \( \sigma \) -algebra. Therefore there are at least one algebra and one \( \sigma \) -algebra containing \( \mathcal{E} \) . We now define \( \alpha \left( \mathcal{E}\right) \) (or \( \sigma \left( \mathcal{E}\right) \) ) to consist of all ... | Yes |
Lemma 2. A necessary and sufficient condition for an algebra \( \mathcal{A} \) to be a \( \sigma \) -algebra is that it is a monotonic class. | Proof. A \( \sigma \) -algebra is evidently a monotonic class. Now let \( \mathcal{A} \) be a monotonic class and \( {A}_{n} \in \mathcal{A}, n = 1,2,\ldots \) It is clear that \( {B}_{n} = \mathop{\bigcup }\limits_{{i = 1}}^{n}{A}_{i} \in \mathcal{A} \) and \( {B}_{n} \subseteq {B}_{n + 1} \) . Consequently, by the de... | Yes |
Any \( \pi - \lambda \) -system \( \mathcal{E} \) is a \( \sigma \) -algebra. | The system \( \mathcal{E} \) contains \( \Omega \) (because of \( \left( {\lambda }_{a}\right) \) ) and is closed under taking complements and finite intersections (because of \( \left( {\lambda }_{b}^{\prime }\right) \) and the assumption that \( \mathcal{E} \) is a \( \pi \) -system). Therefore the system of sets \( ... | Yes |
Lemma 3. Let \( \mathrm{P} \) and \( \mathrm{Q} \) be two probability measures on a measurable space \( \left( {\Omega ,\mathcal{F}}\right) \) . Let \( \mathcal{E} \) be a \( \pi \) -system of sets in \( \mathcal{F} \) and the measures \( \mathrm{P} \) and \( \mathrm{Q} \) coincide on the sets which belong to \( \mathc... | Proof. We will use the principle of appropriate sets taking for these sets \( \mathcal{L} = \) \( \{ A \in \sigma \left( \mathcal{E}\right) : \mathrm{P}\left( A\right) = \mathrm{Q}\left( A\right) \} \) . Clearly, \( \Omega \in \mathcal{L} \) . If \( A \in \mathcal{L} \), then obviously \( \bar{A} \in \mathcal{L} \) , s... | Yes |
Theorem 3. Let \( \mathcal{E} \) be a \( \pi \) -system of sets in \( \mathcal{F} \) and \( \mathcal{H} \) a class of real-valued \( \mathcal{F} \) -measurable functions satisfying the following conditions:\n\n\( \left( {h}_{1}\right) \) if \( A \in \mathcal{E} \), then \( {I}_{A} \in \mathcal{H} \) ;\n\n\( \left( {h}_... | Proof. Let \( \mathcal{L} = \left\{ {A \in \mathcal{F} : {I}_{A} \in \mathcal{H}}\right\} \) . Then \( \left( {h}_{1}\right) \) implies that \( \mathcal{E} \subseteq \mathcal{L} \) . But by \( \left( {h}_{2}\right) \) and \( \left( {h}_{3}\right) \) the system \( \mathcal{L} \) is a \( \lambda \) -system (Problem 11). ... | Yes |
Lemma 5. Let \( \mathcal{E} \) be a class of subsets of \( \Omega \), let \( B \subseteq \Omega \), and define\n\n\[ \mathcal{E} \cap B = \{ A \cap B : A \in \mathcal{E}\} .\n\]\nThen\n\n\[ \sigma \left( {\mathcal{E} \cap B}\right) = \sigma \left( \mathcal{E}\right) \cap B \] | Proof. Since \( \mathcal{E} \subseteq \sigma \left( \mathcal{E}\right) \), we have\n\n\[ \mathcal{E} \cap B \subseteq \sigma \left( \mathcal{E}\right) \cap B \]\n\nBut \( \sigma \left( \mathcal{E}\right) \cap B \) is a \( \sigma \) -algebra; hence it follows from (7) that\n\n\[ \sigma \left( {\mathcal{E} \cap B}\right)... | Yes |
Theorem 4. Let \( T \) be any uncountable set. Then \( \mathcal{B}\left( {R}^{T}\right) = {\mathcal{B}}_{1}\left( {R}^{T}\right) = {\mathcal{B}}_{2}\left( {R}^{T}\right) \), and every set \( A \in \mathcal{B}\left( {R}^{T}\right) \) has the following structure: there are a countable set of points \( {t}_{1},{t}_{2},\ld... | Proof. Let \( \mathcal{E} \) denote the collection of sets of the form (15) (for various aggregates \( \left. {\left( {{t}_{1},{t}_{2},\ldots }\right) \text{and Borel sets}B\text{in}\mathcal{B}\left( {R}^{\infty }\right) }\right) \) . If \( {A}_{1},{A}_{2},\ldots \in \mathcal{E} \) and the corresponding aggregates are ... | Yes |
Theorem 2. Let \( {F}_{n} = {F}_{n}\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a distribution function on \( {R}^{n} \) . Then there is a unique probability measure \( \mathrm{P} \) on \( \left( {{R}^{n},\mathcal{B}\left( {R}^{n}\right) }\right) \) such that\n\n\[ \mathrm{P}(a, b\rbrack = {\Delta }_{{a}_{1}{b}_{1}}\c... | Let \( {F}^{1},\ldots ,{F}^{n} \) be one-dimensional distribution functions (on \( R \) ) and\n\n\[ {F}_{n}\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {F}^{1}\left( {x}_{1}\right) \cdots {F}^{n}\left( {x}_{n}\right) . \]\n\nIt is clear that this function is continuous on the right and satisfies (10) and (11). It is also ... | No |
Theorem 3 (Kolmogorov's Theorem on the Extension of Measures on \( \left( {{R}^{\infty },\mathcal{B}\left( {R}^{\infty }\right) }\right) ) \). Let \( {P}_{1},{P}_{2},\ldots \) be probability measures on \( \left( {R,\mathcal{B}\left( R\right) }\right) ,\left( {{R}^{2},\mathcal{B}\left( {R}^{2}\right) }\right) \) , ... ... | Proof. Let \( {B}^{n} \in \mathcal{B}\left( {R}^{n}\right) \) and let \( {\mathcal{I}}_{n}\left( {B}^{n}\right) \) be the cylinder with base \( {B}^{n} \). We assign the measure \( \mathrm{P}\left( {{\mathcal{I}}_{n}\left( {B}^{n}\right) }\right) \) to this cylinder by taking \( \mathrm{P}\left( {{\mathcal{I}}_{n}\left... | Yes |
Lemma 1. Let \( \mathcal{E} \) be a system of sets such that \( \sigma \left( \mathcal{E}\right) = \mathcal{B}\left( R\right) \) . A necessary and sufficient condition that a function \( \xi = \xi \left( \omega \right) \) is \( \mathcal{F} \) -measurable is that\n\n\[ \n\{ \omega : \xi \left( \omega \right) \in E\} \in... | Proof. The necessity is evident. To prove the sufficiency we again use the principle of appropriate sets (Sect. 2).\n\nLet \( \mathcal{D} \) be the system of those Borel sets \( D \) in \( \mathcal{B}\left( R\right) \) for which \( {\xi }^{-1}\left( D\right) \in \mathcal{F} \) . The operation \ | Yes |
Lemma 2. Let \( \varphi = \varphi \left( x\right) \) be a Borel function and \( \xi = \xi \left( \omega \right) \) a random variable. Then the composition \( \eta = \varphi \circ \xi \), i.e., the function \( \eta \left( \omega \right) = \varphi \left( {\xi \left( \omega \right) }\right) \), is also a random variable. | Proof. This statement follows from the equations\n\n\[ \n\{ \omega : \eta \left( \omega \right) \in B\} = \{ \omega : \varphi \left( {\xi \left( \omega \right) }\right) \in B\} = \left\{ {\omega : \xi \left( \omega \right) \in {\varphi }^{-1}\left( B\right) }\right\} \in \mathcal{F} \n\]\n\n(7)\n\nfor \( B \in \mathcal... | Yes |
Theorem 1. (b) If also \( \xi \left( \omega \right) \geq 0 \), there is a sequence of simple random variables \( {\xi }_{1},{\xi }_{2},\ldots \) such that \( {\xi }_{n}\left( \omega \right) \uparrow \xi \left( \omega \right), n \rightarrow \infty \), for all \( \omega \in \Omega \) . | Proof. We begin by proving the second statement. For \( n = 1,2,\ldots \), put\n\n\[ \n{\xi }_{n}\left( \omega \right) = \mathop{\sum }\limits_{{k = 1}}^{{n{2}^{n}}}\frac{k - 1}{{2}^{n}}{I}_{k, n}\left( \omega \right) + n{I}_{\{ \xi \left( \omega \right) \geq n\} }\left( \omega \right) ,\n\]\n\nwhere \( {I}_{k, n} \) i... | Yes |
Theorem 2. Let \( {\xi }_{1},{\xi }_{2},\ldots \) be a sequence of extended simple random variables and \( \xi \left( \omega \right) = \lim {\xi }_{n}\left( \omega \right) ,\omega \in \Omega \) . Then \( \xi \left( \omega \right) \) is also an extended random variable. | PROOF. It follows immediately from the remark above and the fact that\n\n\[ \n\{ \omega : \xi \left( \omega \right) < x\} = \left\{ {\omega : \lim {\xi }_{n}\left( \omega \right) < x}\right\} \n\]\n\n\[ \n= \left\{ {\omega : \lim \sup {\xi }_{n}\left( \omega \right) = \lim \inf {\xi }_{n}\left( \omega \right) }\right\}... | Yes |
Theorem 3. Let \( \eta = \eta \left( \omega \right) \) be an \( {\mathcal{F}}_{\xi } \) -measurable random variable. Then there is a Borel function \( \varphi \) such that \( \eta = \varphi \circ \xi \), i.e. \( \eta \left( \omega \right) = \varphi \left( {\xi \left( \omega \right) }\right) \) for every \( \omega \in \... | Proof. Let \( \Phi \) be the class of \( {\mathcal{F}}_{\xi } \) -measurable functions \( \eta = \eta \left( \omega \right) \) and \( {\widetilde{\Phi }}_{\xi } \) the class of \( {\mathcal{F}}_{\xi } \) -measurable functions representable in the form \( \varphi \circ \xi \), where \( \varphi \) is a Borel function. It... | Yes |
Lemma 3. Let \( \xi = \xi \left( \omega \right) \) be a \( \sigma \left( \mathcal{D}\right) \) -measurable random variable. Then \( \xi \) is representable in the form\n\n\[ \xi \left( \omega \right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }{x}_{k}{I}_{{D}_{k}}\left( \omega \right) \]\n\n(8)\n\nwhere \( {x}_{k} \in R... | Proof. Let us choose a set \( {D}_{k} \) and show that the \( \sigma \left( \mathcal{D}\right) \) -measurable function \( \xi \) has a constant value on that set. For this purpose, denote\n\n\[ {x}_{k} = \sup \left\lbrack {c : {D}_{k}\cap \{ \omega : \xi \left( \omega \right) < c\} = \varnothing }\right\rbrack .\n\]\n\... | Yes |
Lemma 1. Let \( \eta \) and \( {\xi }_{n} \) be simple nonnegative random variables, \( n \geq 1 \), and\n\n\[ \n{\xi }_{n} \uparrow \xi \geq \eta \n\]\n\nThen\n\n\[ \n\mathop{\lim }\limits_{n}\mathrm{E}{\xi }_{n} \geq \mathrm{E}\eta \n\]\n\n(5) | Proof. Let \( \varepsilon > 0 \) and\n\n\[ \n{A}_{n} = \left\{ {\omega : {\xi }_{n} \geq \eta - \varepsilon }\right\} \n\]\n\nIt is clear that \( {A}_{n} \uparrow \Omega \) and\n\n\[ \n{\xi }_{n} = {\xi }_{n}{I}_{{A}_{n}} + {\xi }_{n}{I}_{{\bar{A}}_{n}} \geq {\xi }_{n}{I}_{{A}_{n}} \geq \left( {\eta - \varepsilon }\rig... | Yes |
Theorem 1 (On Monotone Convergence). Let \( \eta ,\xi ,{\xi }_{1},{\xi }_{2},\ldots \) be random variables.\n\n(a) If \( {\xi }_{n} \geq \eta \) for all \( n \geq 1,\mathrm{E}\eta > - \infty \), and \( {\xi }_{n} \uparrow \xi \), then\n\n\[ \mathrm{E}{\xi }_{n} \uparrow \mathrm{E}\xi \text{.} \]\n\n(b) If \( {\xi }_{n}... | Proof. (a) First suppose that \( \eta \geq 0 \) . For each \( k \geq 1 \) let \( {\left\{ {\xi }_{k}^{\left( n\right) }\right\} }_{n \geq 1} \) be a sequence of simple functions such that \( {\xi }_{k}^{\left( n\right) } \uparrow {\xi }_{k}, n \rightarrow \infty \) . Put \( {\zeta }^{\left( n\right) } = \mathop{\max }\... | Yes |
Theorem 2 (Fatou’s Lemma). Let \( \eta ,{\xi }_{1},{\xi }_{2},\ldots \) be random variables.\n\n(a) If \( {\xi }_{n} \geq \eta \) for all \( n \geq 1 \) and \( \mathrm{E}\eta > - \infty \), then\n\n\[ \mathrm{E}\liminf {\xi }_{n} \leq \liminf \mathrm{E}{\xi }_{n} \] | Proof. (a) Let \( {\zeta }_{n} = \mathop{\inf }\limits_{{m \geq n}}{\xi }_{m} \) ; then\n\n\[ \lim \inf {\xi }_{n} = \mathop{\lim }\limits_{n}\mathop{\inf }\limits_{{m \geq n}}{\xi }_{m} = \mathop{\lim }\limits_{n}{\zeta }_{n} \]\n\nIt is clear that \( {\zeta }_{n} \uparrow \liminf {\xi }_{n} \) and \( {\zeta }_{n} \ge... | Yes |
Theorem 3 (Lebesgue’s Theorem on Dominated Convergence). Let \( \eta ,\xi ,{\xi }_{1},{\xi }_{2},\ldots \) be random variables such that \( \left| {\xi }_{n}\right| \leq \eta ,\mathrm{E}\eta < \infty \) and \( {\xi }_{n} \rightarrow \xi \) (a.s.). Then \( \mathrm{E}\left| \xi \right| < \) 00, \[ \mathrm{E}{\xi }_{n} \r... | Proof. By hypothesis, \( \lim \inf {\xi }_{n} = \lim \sup {\xi }_{n} = \xi \) (a.s.). Therefore by Property \( \mathbf{G} \) and Fatou's lemma (item (c)) \[ \mathbf{E}\xi = \mathbf{E}\liminf {\xi }_{n} \leq \liminf \mathbf{E}{\xi }_{n} = \limsup \mathbf{E}{\xi }_{n} = \mathbf{E}\limsup {\xi }_{n} = \mathbf{E}\xi \] whi... | Yes |
Theorem 4. Let \( {\left\{ {\xi }_{n}\right\} }_{n \geq 1} \) be a uniformly integrable family of random variables. Then\n\n(a) \( \mathrm{E}\liminf {\xi }_{n} \leq \liminf \mathrm{E}{\xi }_{n} \leq \limsup \mathrm{E}{\xi }_{n} \leq \mathrm{E}\limsup {\xi }_{n} \) . | Proof. (a) For every \( c > 0 \)\n\n\[ \mathrm{E}{\xi }_{n} = \mathrm{E}\left\lbrack {{\xi }_{n}{I}_{\left\{ {\xi }_{n} < - c\right\} }}\right\rbrack + \mathrm{E}\left\lbrack {{\xi }_{n}{I}_{\left\{ {\xi }_{n} \geq - c\right\} }}\right\rbrack .\n\]\n\n(12)\n\nBy uniform integrability, for every \( \varepsilon > 0 \) we... | Yes |
Theorem 5. Let \( 0 \leq {\xi }_{n} \rightarrow \xi \) (P- a.s.) and \( \mathrm{E}{\xi }_{n} < \infty, n \geq 1 \) . Then \( \mathrm{E}{\xi }_{n} \rightarrow \mathrm{E}\xi < \infty \) if and only if the family \( {\left\{ {\xi }_{n}\right\} }_{n \geq 1} \) is uniformly integrable. | Proof. The sufficiency follows from conclusion (b) of Theorem 4. For the proof of the necessity we consider the (at most countable) set\n\n\[ A = \{ a : \mathrm{P}\left( {\xi = a}\right) > 0\} . \]\n\nThen we have \( {\xi }_{n}{I}_{\left\{ {\xi }_{n} < a\right\} } \rightarrow \xi {I}_{\{ \xi < a\} } \) for each \( a \n... | Yes |
A necessary and sufficient condition for a family \( {\left\{ {\xi }_{n}\right\} }_{n \geq 1} \) of random variables to be uniformly integrable is that \( \mathrm{E}\left| {\xi }_{n}\right|, n \geq 1 \), are uniformly bounded (i.e.,(16) holds) and that \( \mathrm{E}\left\{ {\left| {\xi }_{n}\right| {I}_{A}}\right\}, n ... | Necessity. Condition (16) was verified above. Moreover,\n\n\[ \mathrm{E}\left\{ {\left| {\xi }_{n}\right| {I}_{A}}\right\} = \mathrm{E}\left\{ {\left| {\xi }_{n}\right| {I}_{A\cap \{ \left| {\xi }_{n}\right| \geq c\} }}\right\} + \mathrm{E}\left\{ {\left| {\xi }_{n}\right| {I}_{A\cap \{ \left| {\xi }_{n}\right| < c\} }... | Yes |
Lemma 3. Let \( {\xi }_{1},{\xi }_{2},\ldots \) be a sequence of integrable random variables and \( G = \) \( G\left( t\right) \) a nonnegative increasing function, defined for \( t \geq 0 \), such that\n\n\[ \mathop{\lim }\limits_{{t \rightarrow \infty }}\frac{G\left( t\right) }{t} = \infty \]\n\n(18)\n\n\[ \mathop{\s... | Proof. Let \( \varepsilon > 0, M = \mathop{\sup }\limits_{n}\mathrm{E}\left\lbrack {G\left( \left| {\xi }_{n}\right| \right) }\right\rbrack, a = M/\varepsilon \) . Take \( c \) so large that \( G\left( t\right) /t \geq a \) for \( t \geq c \) . Then\n\n\[ \mathrm{E}\left\lbrack {\left| {\xi }_{n}\right| {I}_{\left\{ \l... | Yes |
Theorem 6. Let \( \xi \) and \( \eta \) be independent random variables, \( \mathrm{E}\left| \xi \right| < \infty ,\mathrm{E}\left| \eta \right| < \infty \) . Then \( \mathrm{E}\left| {\xi \eta }\right| < \infty \) and\n\n\[ \mathbf{E}{\xi \eta } = \mathbf{E}\xi \cdot \mathbf{E}\eta \] | Proof. First let \( \xi \geq 0,\eta \geq 0 \) . Put\n\n\[ {\xi }_{n} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{k}{n}{I}_{\{ k/n \leq \xi \left( \omega \right) < \left( {k + 1}\right) /n\} },\]\n\n\[ {\eta }_{n} = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{k}{n}{I}_{\{ k/n \leq \eta \left( \omega \right) < \l... | Yes |
Theorem 7 (Change of Variables in a Lebesgue Integral). Let \( \left( {\Omega ,\mathcal{F}}\right) \) and \( \left( {E,\mathcal{E}}\right) \) be measurable spaces and \( X = X\left( \omega \right) \) an \( \mathcal{F}/\mathcal{E} \) -measurable function with values in E. Let \( \mathrm{P} \) be a probability measure on... | Proof. Let \( A \in \mathcal{E} \) and \( g\left( x\right) = {I}_{B}\left( x\right) \), where \( B \in \mathcal{E} \) . Then (41) becomes\n\n\[ \n{P}_{X}\left( {AB}\right) = \mathrm{P}\left( {{X}^{-1}\left( A\right) \cap {X}^{-1}\left( B\right) }\right)\n\]\n\n(42)\n\nwhich follows from (40) and the observation that \(... | Yes |
Theorem 8 (Fubini’s Theorem). Let \( \xi = \xi \left( {{\omega }_{1},{\omega }_{2}}\right) \) be an \( {\mathcal{F}}_{1} \otimes {\mathcal{F}}_{2} \) -measurable function, integrable with respect to the measure \( {\rho }_{1} \times {\rho }_{2} \) :\n\n\[ \n{\int }_{{\Omega }_{1} \times {\Omega }_{2}}\left| {\xi \left(... | Proof. We first show that \( {\xi }_{{\omega }_{1}}\left( {\omega }_{2}\right) = \xi \left( {{\omega }_{1},{\omega }_{2}}\right) \) is \( {\mathcal{F}}_{2} \) -measurable with respect to \( {\omega }_{2} \), for each \( {\omega }_{1} \in {\Omega }_{1} \). \n\nLet \( F \in {\mathcal{F}}_{1} \otimes {\mathcal{F}}_{2} \) ... | Yes |
Theorem 9. If \( g = g\left( x\right) \) is continuous on \( \left\lbrack {a, b}\right\rbrack \), it is Riemann-Stieltjes integrable and \[ \left( {\mathrm{R} - \mathrm{S}}\right) {\int }_{a}^{b}g\left( x\right) G\left( {dx}\right) = \left( {\mathrm{L} - \mathrm{S}}\right) {\int }_{a}^{b}g\left( x\right) G\left( {dx}\r... | Proof. Since \( g\left( x\right) \) is continuous, we have \( \bar{g}\left( x\right) = g\left( x\right) = \underline{g}\left( x\right) \) . Hence by (57) \( \mathop{\lim }\limits_{{k \rightarrow \infty }}{\mathop{\sum }\limits_{{\mathcal{P}}_{k}}}_{ = } = \mathop{\lim }\limits_{{k \rightarrow \infty }}\mathop{\sum }\li... | Yes |
Theorem 10. Let \( g\left( x\right) \) be a bounded function on \( \left\lbrack {a, b}\right\rbrack \) .\n\n(a) The function \( g = g\left( x\right) \) is Riemann integrable on \( \left\lbrack {a, b}\right\rbrack \) if and only if it is continuous almost everywhere (with respect to Lebesgue measure \( \bar{\lambda } \)... | Proof. (a) Let \( g = g\left( x\right) \) be Riemann integrable. Then, by (57),\n\n\[ \text{(L)}{\int }_{a}^{b}\bar{g}\left( x\right) \bar{\lambda }\left( {dx}\right) = \text{(L)}{\int }_{a}^{b}\underline{g}\left( x\right) \bar{\lambda }\left( {dx}\right) \text{.} \]\n\nBut \( \underline{g}\left( x\right) \leq g\left( ... | Yes |
Theorem 11. The following formulas are valid for all real a and \( b, a < b \) :\n\n\[ F\left( b\right) G\left( b\right) - F\left( a\right) G\left( a\right) = {\int }_{a}^{b}F\left( {s - }\right) {dG}\left( s\right) + {\int }_{a}^{b}G\left( s\right) {dF}\left( s\right) ,\]\n\n(62)\n\nor equivalently\n\n\[ F\left( b\rig... | Proof. We first recall that in accordance with Subsection 1 an integral \( {\int }_{a}^{b}\left( \cdot \right) \) means \( {\int }_{(a, b\rbrack }\left( \cdot \right) \). Then (see formula (2) in Sect. 3)\n\n\[ \left( {F\left( b\right) - F\left( a\right) }\right) \left( {G\left( b\right) - G\left( a\right) }\right) = {... | Yes |
If \( F\left( x\right) \) and \( G\left( x\right) \) are distribution functions, then\n\n\[ F\left( x\right) G\left( x\right) = {\int }_{-\infty }^{x}F\left( {s - }\right) {dG}\left( s\right) + {\int }_{-\infty }^{x}G\left( s\right) {dF}\left( s\right) . \] | If also\n\n\[ F\left( x\right) = {\int }_{-\infty }^{x}f\left( s\right) {ds} \]\n\nthen\n\n\[ F\left( x\right) G\left( x\right) = {\int }_{-\infty }^{x}F\left( s\right) {dG}\left( s\right) + {\int }_{-\infty }^{x}G\left( s\right) f\left( s\right) {ds}. \] | No |
Corollary 2. Let \( \xi \) be a random variable with distribution function \( F\left( x\right) \) and E \( {\left| \xi \right| }^{n} < \infty \) . Then\n\n\[ \n{\int }_{0}^{\infty }{x}^{n}{dF}\left( x\right) = n{\int }_{0}^{\infty }{x}^{n - 1}\left\lbrack {1 - F\left( x\right) }\right\rbrack {dx} \]\n\n(69)\n\n\[ \n{\i... | To prove (69) we observe that\n\n\[ \n{\int }_{0}^{b}{x}^{n}{dF}\left( x\right) = - {\int }_{0}^{b}{x}^{n}d\left( {1 - F\left( x\right) }\right) \]\n\n\[ \n= - {b}^{n}\left( {1 - F\left( b\right) }\right) + n{\int }_{0}^{b}{x}^{n - 1}\left( {1 - F\left( x\right) }\right) {dx}. \]\n\n(72)\n\nLet us show that since \( \m... | Yes |
Theorem 1. If \( \mathcal{G} = \sigma \left( \mathcal{D}\right) \) and \( \xi \) is a random variable for which \( \mathrm{E}\xi \) is defined, then\n\n\[ \n\mathrm{E}\left( {\xi \mid \mathcal{G}}\right) = \mathrm{E}\left( {\xi \mid {D}_{i}}\right) \;\left( {\mathrm{P}\text{-a.s. on }{D}_{i}}\right)\n\]\n\nor equivalen... | Proof. According to Lemma 3 of Sect. 4, \( \mathrm{E}\left( {\xi \mid \mathcal{G}}\right) = {K}_{i} \) on \( {D}_{i} \), where \( {K}_{i} \) are constants. But\n\n\[ \n{\int }_{{D}_{i}}{\xi d}\mathrm{P} = {\int }_{{D}_{i}}\mathrm{E}\left( {\xi \mid \mathcal{G}}\right) d\mathrm{P} = {K}_{i}\mathrm{P}\left( {D}_{i}\right... | Yes |
Theorem 2 (On Taking Limits Under the Conditional Expectation Sign). Let \( {\left\{ {\xi }_{n}\right\} }_{n \geq 1} \) be a sequence of extended random variables.\n\n(a) If \( \left| {\xi }_{n}\right| \leq \eta ,\mathrm{E}\eta < \infty \), and \( {\xi }_{n} \rightarrow \xi \) (a.s.), then\n\n\[ \mathrm{E}\left( {{\xi ... | Proof. (a) Let \( {\zeta }_{n} = \mathop{\sup }\limits_{{m > n}}\left| {{\xi }_{m} - \xi }\right| \) . Since \( {\xi }_{n} \rightarrow \xi \) (a. s.), we have \( {\zeta }_{n} \downarrow 0 \) (a. s.). The expectations \( \mathrm{E}{\xi }_{n} \) and \( \overline{\mathrm{E}}\xi \) are finite; therefore by Properties D* an... | Yes |
Let \( \eta \) be a discrete random variable with \( \mathrm{P}\left( {\eta = {y}_{k}}\right) > 0,\mathop{\sum }\limits_{{k = 1}}^{\infty }\mathrm{P}(\eta = \) \( \left. {y}_{k}\right) = 1 \) . Then | \[ \mathrm{P}\left( {A \mid \eta = {y}_{k}}\right) = \frac{\mathrm{P}\left( {A \cap \left\{ {\eta = {y}_{k}}\right\} }\right) }{\mathrm{P}\left( {\eta = {y}_{k}}\right) },\;k \geq 1. \] For \( y \notin \left\{ {{y}_{1},{y}_{2},\ldots }\right\} \) the conditional probability \( \mathrm{P}\left( {A \mid \eta = y}\right) ... | Yes |
Let \( \left( {\xi ,\eta }\right) \) be a pair of random variables whose distribution has a density \( {f}_{\xi ,\eta }\left( {x, y}\right) \). Let \( {f}_{\xi }\left( x\right) \) and \( {f}_{\eta }\left( y\right) \) be the densities of the probability distributions of \( \xi \) and \( \eta \). Let us put \( {f}_{\xi \... | In fact, to prove (19) it is enough to verify (17) for \( B \in \mathcal{B}\left( R\right), A = \{ \xi \in C\} \). By (43) and (45) of Sect. 6 and Fubini's theorem, \[ {\int }_{B}\left\lbrack {{\int }_{C}{f}_{\xi \mid \eta }\left( {x \mid y}\right) {dx}}\right\rbrack {P}_{\eta }\left( {dy}\right) = {\int }_{B}\left\lbr... | Yes |
Example 3. Let the length of time that a piece of apparatus will continue to operate be described by a nonnegative random variable \( \eta = \eta \left( \omega \right) \) whose distribution function \( {F}_{\eta }\left( y\right) \) has a density \( {f}_{\eta }\left( y\right) \) (naturally, \( {F}_{\eta }\left( y\right)... | Let \( \mathrm{P}\left( {\eta \geq a}\right) > 0 \) . Then according to the definition of conditional probability given in Subsection 1 and (45) of Sect. 6 we have\n\n\[ \mathrm{E}\left( {\eta - a \mid \eta \geq a}\right) = \frac{\mathrm{E}\left\lbrack {\left( {\eta - a}\right) {I}_{\{ \eta \geq a\} }}\right\rbrack }{\... | Yes |
Theorem 3. Let \( P\left( {\omega ;B}\right) \) be a regular conditional probability with respect to \( \mathcal{G} \) and let \( \xi \) be an integrable random variable. Then\n\n\[ \mathrm{E}\left( {\xi \mid \mathcal{G}}\right) \left( \omega \right) = {\int }_{\Omega }\xi \left( \widetilde{\omega }\right) P\left( {\om... | Proof. If \( \xi = {I}_{B}, B \in \mathcal{F} \), the required formula (24) becomes\n\n\[ \mathrm{P}\left( {B \mid \mathcal{G}}\right) \left( \omega \right) = P\left( {\omega ;B}\right) \;\text{ (a. s.),} \]\n\nwhich holds by Definition 6 (b). Consequently (24) holds for simple functions.\n\nNow let \( \xi \geq 0 \) an... | Yes |
Theorem 5. Let \( X = X\\left( \\omega \\right) \) be a random element with values in the Borel space \( \\left( {E,\\mathcal{E}}\\right) \) . Then there is a regular conditional distribution of \( X \) with respect to \( \\mathcal{G} \\subseteq \\mathcal{F} \) . | Proof. Let \( \\varphi = \\varphi \\left( e\\right) \) be the function in Definition 9. By (2) in this definition \( \\varphi \\left( {X\\left( \\omega \\right) }\\right) \) is a random variable. Hence, by Theorem 4, we can define the conditional distribution \( Q\\left( {\\omega ;A}\\right) \) of \( \\varphi \\left( {... | Yes |
Theorem 7. Let \( \mathcal{P} = \left\{ {{\mathrm{P}}_{\theta },\theta \in \Theta }\right\} \) be a dominated family, i.e., there exists a \( \sigma \) - finite measure \( \lambda \) on \( \left( {\Omega ,\mathcal{F}}\right) \) such that the measures \( {\mathrm{P}}_{\theta } \) are absolutely continuous with respect t... | Proof. Sufficiency. By assumption, the dominating measure \( \lambda \) is \( \sigma \) -finite. This means that there are \( \mathcal{F} \) -measurable disjoint sets \( {\Omega }_{k}, k \geq 1 \), such that \( \Omega = \mathop{\sum }\limits_{{k \geq 1}}{\Omega }_{k} \) and \( 0 < \lambda \left( {\Omega }_{k}\right) < ... | Yes |
Example 5 (Exponential Family). Assume that \( \Omega = {R}^{n},\mathcal{F} = \mathcal{B}\left( {R}^{n}\right) \) and the measure \( {\mathrm{P}}_{\theta } \) is such that\n\n\[ \n{P}_{\theta }\left( {d\omega }\right) = {P}_{\theta }\left( {d{x}_{1}}\right) \cdots {P}_{\theta }\left( {d{x}_{n}}\right) \n\]\n\n(54)\n\nf... | It follows from (54) and (55) that\n\n\[ \n{\mathrm{P}}_{\theta }\left( {d\omega }\right) = {\alpha }^{n}\left( \theta \right) {e}^{\beta \left( \theta \right) \left\lbrack {s\left( {x}_{1}\right) + \cdots + s\left( {x}_{n}\right) }\right\rbrack }\gamma \left( {x}_{1}\right) \cdots \gamma \left( {x}_{n}\right) d{x}_{1}... | Yes |
Theorem 1. Let \( \mathrm{E}{\eta }^{2} < \infty \) . Then there is an optimal estimator \( {\varphi }^{ * } = {\varphi }^{ * }\left( \xi \right) \) and \( {\varphi }^{ * }\left( x\right) \) can be taken to be the function\n\n\[ \n{\varphi }^{ * }\left( x\right) = \mathrm{E}\left( {\eta \mid \xi = x}\right) .\n\] | Proof. Without loss of generality we may consider only estimators \( \varphi \left( \xi \right) \) for which \( \mathrm{E}{\varphi }^{2}\left( \xi \right) < \infty \) . Then if \( \varphi \left( \xi \right) \) is such an estimator, and \( {\varphi }^{ * }\left( \xi \right) = \mathrm{E}\left( {\eta \mid \xi }\right) \),... | Yes |
Theorem 2 (Theorem on the Normal Correlation). Let \( \left( {\xi ,\eta }\right) \) be a Gaussian vector with \( \operatorname{Var}\xi > 0 \) . Then the optimal estimator of \( \eta \) in terms of \( \xi \) is\n\n\[ \mathrm{E}\left( {\eta \mid \xi }\right) = \mathrm{E}\eta + \frac{\operatorname{Cov}\left( {\xi ,\eta }\... | and its error is\n\n\[ \Delta \equiv \mathrm{E}{\left\lbrack \eta - \mathrm{E}\left( \eta \mid \xi \right) \right\rbrack }^{2} = \operatorname{Var}\eta - \frac{{\operatorname{Cov}}^{2}\left( {\xi ,\eta }\right) }{\operatorname{Var}\xi }. \] | Yes |
Theorem 1 (Kolmogorov's Theorem on the Existence of a Process). Let \( \\left\\{ {{F}_{{t}_{1},\\ldots ,{t}_{n}}\\left( {{x}_{1},\\ldots ,{x}_{n}}\\right) }\\right\\} \), with \( {t}_{i} \\in T \\subseteq R,{t}_{1} < {t}_{2} < \\cdots < {t}_{n}, n \\geq 1 \), be a given family of finite-dimensional distribution functio... | Proof. Put\n\n\[ \n\\Omega = {R}^{T},\\;\\mathcal{F} = \\mathcal{B}\\left( {R}^{T}\\right) \n\]\n\ni.e., take \( \\Omega \) to be the space of real functions \( \\omega = {\\left( {\\omega }_{t}\\right) }_{t \\in T} \) with the \( \\sigma \) -algebra generated by the cylindrical sets.\n\nLet \( \\tau = \\left\\lbrack {... | Yes |
Let \( {F}_{1}\left( x\right) ,{F}_{2}\left( x\right) ,\ldots \) be a sequence of one-dimensional distribution functions. Then there exist a probability space \( \left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) and a sequence of independent random variables \( {\xi }_{1},{\xi }_{2},\ldots \) such that\n\n\[ \mathrm{P}... | To establish the corollary it is enough to put \( {F}_{1,\ldots, n}\left( {{x}_{1},\ldots ,{x}_{n}}\right) = {F}_{1}\left( {x}_{1}\right) \cdots \) \( {F}_{n}\left( {x}_{n}\right) \) and apply Theorem 1. | No |
Let \( T = \lbrack 0,\infty ) \) and let \( \{ P(s, x;t, B\} \) be a family of nonnegative functions defined for \( s, t \in T, t > s, x \in R, B \in \mathcal{B}\left( R\right) \), and satisfying the following conditions:\n\n(a) \( P\left( {s, x;t, B}\right) \) is a probability measure in \( B \) for given \( s, x \) a... | The process \( X \) so constructed is a Markov process with initial distribution \( \pi \) and transition probabilities \( \{ P(s, x;t, B\} \) . | Yes |
Theorem 2 (Ionescu Tulcea's Theorem on Extending a Measure and the Existence of a Random Sequence). Let \( \left( {{\Omega }_{n},{\mathcal{F}}_{n}}\right), n = 1,2,\ldots \), be arbitrary measurable spaces and \( \Omega = \prod {\Omega }_{n},\mathcal{F} = {\overline{\lbrack s\rbrack }}_{n} \) . Suppose that a probabili... | Proof. The first step is to establish that for each \( n > 1 \) the set function \( {P}_{n} \) defined by (9) on the rectangles \( {A}_{1} \times \cdots \times {A}_{n} \) can be extended to the \( \sigma \) -algebra \( {\mathcal{F}}^{n} \) .\n\nFor each \( n \geq 2 \) and \( B \in {\mathcal{F}}^{n} \) we put\n\n\[ {P}_... | Yes |
Corollary 4. Let \( {\left( {E}_{n},{\mathcal{E}}_{n}\right) }_{n \geq 1} \) be any measurable spaces and \( {\left( {P}_{n}\right) }_{n \geq 1} \) measures on them. Then there are a probability space \( \left( {\Omega ,\mathcal{F},\mathbf{P}}\right) \) and a family of independent random elements \( {X}_{1},{X}_{2},\ld... | \[ \mathrm{P}\left\{ {\omega : {X}_{n}\left( \omega \right) \in B}\right\} = {P}_{n}\left( B\right) ,\;B \in {\mathcal{E}}_{n}, n \geq 1. \] | No |
Corollary 5. Let \( E = \{ 1,2,\ldots \} \), and let \( \left\{ {{p}_{k}\left( {x, y}\right) }\right\} \) be a family of nonnegative functions, \( k \geq 1, x, y \in E \), such that \( \mathop{\sum }\limits_{{y \in E}}{p}_{k}\left( {x;y}\right) = 1, x \in E, k \geq 1 \) . Also let \( \pi = \pi \left( x\right) \) be a p... | Then there are a probability space \( \left( {\Omega ,\mathcal{F},\mathrm{P}}\right) \) and a family \( X = \left\{ {{\xi }_{0},{\xi }_{1},\ldots }\right\} \) of random variables on it such that\n\n\[ \mathrm{P}\left\{ {{\xi }_{0} = {x}_{0},{\xi }_{1} = {x}_{1},\ldots ,{\xi }_{n} = {x}_{n}}\right\} = \pi \left( {x}_{0}... | Yes |
If \( {A}_{n}^{\varepsilon } = \left\{ {\omega : \left| {{\xi }_{n} - \xi }\right| \geq \varepsilon }\right\} \) then (8) means that \( \mathop{\sum }\limits_{{n = 1}}^{\infty }\mathrm{P}\left( {A}_{n}^{\varepsilon }\right) < \infty \) , \( \varepsilon > 0 \), and then by the Borel-Cantelli lemma we have \( \mathrm{P}\... | \[ \mathop{\sum }\limits_{{k = 1}}^{\infty }\mathrm{P}\left\{ {\left| {{\xi }_{k} - \xi }\right| \geq \varepsilon }\right\} < \infty ,\varepsilon > 0 \Rightarrow \mathrm{P}\left( {A}^{\varepsilon }\right) = 0,\varepsilon > 0 \] \[ \Leftrightarrow \mathrm{P}\left\{ {\omega : {\xi }_{n} \nrightarrow \xi )}\right\} = 0 \]... | Yes |
Example 2. \( \left( {{\xi }_{n}\overset{\mathrm{P}}{ \rightarrow }\xi \Leftarrow {\xi }_{n}\overset{\text{ a.s. }}{ \rightarrow }\xi \nRightarrow {\xi }_{n}\overset{{L}^{p}}{ \rightarrow }\xi, p > 0}\right. \) .) Again let \( \Omega = \left\lbrack {0,1}\right\rbrack ,\mathcal{F} = \) \( \mathcal{B}\left\lbrack {0,1}\r... | \[ {\xi }_{n}\left( \omega \right) = \left\{ \begin{array}{ll} {e}^{n}, & 0 \leq \omega \leq 1/n \\ 0, & \omega > 1/n \end{array}\right. \] Then \( \left\{ {\xi }_{n}\right\} \) converges with probability 1 (and therefore in probability) to zero, but \[ \mathrm{E}{\left| {\xi }_{n}\right| }^{p} = \frac{{e}^{np}}{n} \ri... | Yes |
Example 3. \( \\left( {{\\xi }_{n}\\overset{{L}^{p}}{ \\rightarrow }\\xi \\nRightarrow {\\xi }_{n}\\overset{\\text{ a.s. }}{ \\rightarrow }\\xi \\text{.) Let }\\left\\{ {\\xi }_{n}\\right\\} \\text{be a sequence of independent random}}\\right) \\) variables with | \[ \\mathrm{P}\\left( {{\\xi }_{n} = 1}\\right) = {p}_{n},\\;\\mathrm{P}\\left( {{\\xi }_{n} = 0}\\right) = 1 - {p}_{n}. \]\n\nThen it is easy to show that\n\n\[ {\\xi }_{n}\\overset{\\mathrm{P}}{ \\rightarrow }0 \\Leftrightarrow {p}_{n} \\rightarrow 0,\\;n \\rightarrow \\infty , \]\n\n(14)\n\n\[ {\\xi }_{n}\\overset{{... | Yes |
Theorem 3. Let \( \\left( {\\xi }_{n}\\right) \) be a sequence of nonnegative random variables such that \( {\\xi }_{n}\\overset{\\text{ a. s. }}{ \\rightarrow }\\xi \) and \( \\mathrm{E}{\\xi }_{n} \\rightarrow \\mathrm{E}\\xi < \\infty \) . Then\n\n\[ \n\\mathrm{E}\\left| {{\\xi }_{n} - \\xi }\\right| \\rightarrow 0,... | Proof. We have \( \\mathrm{E}{\\xi }_{n} < \\infty \) for sufficiently large \( n \), and therefore for such \( n \) we have\n\n\[ \n\\mathrm{E}\\left| {\\xi - {\\xi }_{n}}\\right| = \\mathrm{E}\\left( {\\xi - {\\xi }_{n}}\\right) {I}_{\\left\\{ \\xi \\geq {\\xi }_{n}\\right\\} } + \\mathrm{E}\\left( {{\\xi }_{n} - \\x... | Yes |
Theorem 4 (Cauchy Criterion for Almost Sure Convergence). A necessary and sufficient condition for the sequence \( {\left( {\xi }_{n}\right) }_{n \geq 1} \) of random variables to converge with probability 1 (to a random variable \( \xi \) ) is that it is fundamental with probability 1. | Proof. If \( {\xi }_{n}\overset{\text{ a. s. }}{ \rightarrow }\xi \) then\n\n\[ \mathop{\sup }\limits_{\substack{{k \geq n} \\ {l \geq n} }}\left| {{\xi }_{k} - {\xi }_{l}}\right| \leq \mathop{\sup }\limits_{{k \geq n}}\left| {{\xi }_{k} - \xi }\right| + \mathop{\sup }\limits_{{l \geq n}}\left| {{\xi }_{l} - \xi }\righ... | Yes |
Theorem 5. If the sequence \( \left( {\xi }_{n}\right) \) is fundamental (or convergent) in probability, it contains a subsequence \( \left( {\xi }_{{n}_{k}}\right) \) that is fundamental (or convergent) with probability 1 . | Proof. Let \( \left( {\xi }_{n}\right) \) be fundamental in probability. By Theorem 4 it is enough to show that it contains a subsequence that converges almost surely.\n\nTake \( {n}_{1} = 1 \) and define \( {n}_{k} \) inductively as the smallest \( n > {n}_{k - 1} \) for which\n\n\[ \mathrm{P}\left\{ {\left| {{\xi }_{... | Yes |
Theorem 6 (Cauchy Criterion for Convergence in Probability). A necessary and sufficient condition for a sequence \( {\left( {\xi }_{n}\right) }_{n \geq 1} \) of random variables to converge in probability is that it is fundamental in probability. | Proof. If \( {\xi }_{n}\overset{\mathrm{P}}{ \rightarrow }\xi \) then\n\n\[ \mathbf{P}\left\{ {\left| {{\xi }_{n} - {\xi }_{m}}\right| \geq \varepsilon }\right\} \leq \mathbf{P}\left\{ {\left| {{\xi }_{n} - \xi }\right| \geq \varepsilon /2}\right\} + \mathbf{P}\left\{ {\left| {{\xi }_{m} - \xi }\right| \geq \varepsilon... | Yes |
Theorem 7 (Cauchy Test for Convergence in the \( p \) th Mean). A necessary and sufficient condition that a sequence \( {\left( {\xi }_{n}\right) }_{n \geq 1} \) of random variables in \( {L}^{p} \) converges in the mean of order \( p \) to a random variable in \( {L}^{p} \) is that the sequence is fundamental in the m... | Proof. The necessity follows from Minkowski’s inequality. Let \( \left( {\xi }_{n}\right) \) be fundamental \( \left( {{\begin{Vmatrix}{\xi }_{n} - {\xi }_{m}\end{Vmatrix}}_{p} \rightarrow 0, n, m \rightarrow \infty }\right) \) . As in the proof of Theorem 5, we select a subsequence \( \left( {\xi }_{{n}_{k}}\right) \)... | Yes |
Let \( \Omega = R,\mathcal{F} = \mathcal{B}\left( R\right) \), and let \( \mathrm{P} \) be the Gaussian measure,\n\n\[ \mathrm{P}( - \infty, a\rbrack = {\int }_{-\infty }^{a}\varphi \left( x\right) {dx},\;\varphi \left( x\right) = \frac{1}{\sqrt{2\pi }}{e}^{-{x}^{2}/2}. \]\n\nLet \( D = d/{dx} \) and\n\n\[ {H}_{n}\left... | We find easily that\n\n\[ {D\varphi }\left( x\right) = - {x\varphi }\left( x\right) \]\n\n\[ {D}^{2}\varphi \left( x\right) = \left( {{x}^{2} - 1}\right) \varphi \left( x\right) ,\]\n\n\[ {D}^{3}\varphi \left( x\right) = \left( {{3x} - {x}^{3}}\right) \varphi \left( x\right) \]\n\nIt follows that \( {H}_{n}\left( x\rig... | Yes |
Let \( \Omega = \{ 0,1,2,\ldots \} \) and let \( P = \left\{ {{P}_{1},{P}_{2},\ldots }\right\} \) be the Poisson distribution\n\n\[ \n{P}_{x} = \frac{{e}^{-\lambda }{\lambda }^{x}}{x!},\;x = 0,1,\ldots ;\;\lambda > 0.\n\]\n\nPut \( {\Delta f}\left( x\right) = f\left( x\right) - f\left( {x - 1}\right) \left( {f\left( x\... | Since\n\n\[ \n\left( {{\Pi }_{m},{\Pi }_{n}}\right) = \mathop{\sum }\limits_{{x = 0}}^{\infty }{\Pi }_{m}\left( x\right) {\Pi }_{n}\left( x\right) {P}_{x} = {c}_{n}{\delta }_{mn},\n\]\n\nwhere \( {c}_{n} \) are positive constants, the system of normalized Poisson-Charlier polynomials \( {\left\{ {\pi }_{n}\left( x\righ... | Yes |
We define random variables \( {\xi }_{1}\left( x\right) ,{\xi }_{2}\left( x\right) ,\ldots \) by putting \( {\xi }_{n}\left( x\right) = {x}_{n} \). Then for any numbers \( {a}_{i} \), equal to 0 or 1, \( \mathrm{P}\left\{ {x : {\xi }_{1} = {a}_{1},\ldots ,{\xi }_{n} = {a}_{n}}\right\} \) | \[ \mathrm{P}\left\{ {x : {\xi }_{1} = {a}_{1},\ldots ,{\xi }_{n} = {a}_{n}}\right\} = \mathrm{P}\left\{ {x : \frac{{a}_{1}}{2} + \frac{{a}_{2}}{{2}^{2}} + \cdots + \frac{{a}_{n}}{{2}^{n}} \leq x < \frac{{a}_{1}}{2} + \frac{{a}_{2}}{{2}^{2}} + \cdots + \frac{{a}_{n}}{{2}^{n}} + \frac{1}{{2}^{n}}}\right\} = \mathrm{P}\l... | Yes |
The characteristic function of a Bernoulli random variable \( \xi \) with \( \mathrm{P}\left( {\xi = 1}\right) = p,\mathrm{P}\left( {\xi = 0}\right) = q, p + q = 1,1 > p > 0 \), is | \[ {\varphi }_{\xi }\left( t\right) = p{e}^{it} + q \] | Yes |
Let \( \xi \sim \mathcal{N}\left( {m,{\sigma }^{2}}\right) ,\left| m\right| < \infty ,{\sigma }^{2} > 0 \) . Let us show that\n\n\[{\varphi }_{\xi }\left( t\right) = {e}^{{itm} - {t}^{2}{\sigma }^{2}/2}.\] | Let \( \eta = \left( {\xi - m}\right) /\sigma \) . Then \( \eta \sim \mathcal{N}\left( {0,1}\right) \) and, since\n\n\[{\varphi }_{\xi }\left( t\right) = {e}^{itm}{\varphi }_{\eta }\left( {\sigma t}\right)\]\n\nby (5), it is enough to show that\n\n\[{\varphi }_{\eta }\left( t\right) = {e}^{-{t}^{2}/2}\]\n\nWe have\n\n\... | Yes |
Let \( \xi \) be a Poisson random variable,\n\n\[ \n\\mathrm{P}\\left( {\\xi = k}\\right) = \\frac{{e}^{-\\lambda }{\\lambda }^{k}}{k!},\\;k = 0,1,\\ldots \n\]\n\nThen\n\n\[ \n\\mathrm{E}{e}^{it\\xi } = \\mathop{\\sum }\\limits_{{k = 0}}^{\\infty }{e}^{itk}\\frac{{e}^{-\\lambda }{\\lambda }^{k}}{k!} = {e}^{-\\lambda }\... | \[ \n\\mathrm{E}{e}^{it\\xi } = \\mathop{\\sum }\\limits_{{k = 0}}^{\\infty }{e}^{itk}\\frac{{e}^{-\\lambda }{\\lambda }^{k}}{k!} = {e}^{-\\lambda }\\mathop{\\sum }\\limits_{{k = 0}}^{\\infty }\\frac{{\\left( \\lambda {e}^{it}\\right) }^{k}}{k!} = \\exp \\left\\{ {\\lambda \\left( {{e}^{it} - 1}\\right) }\\right\\} .\n... | Yes |
Theorem 1. Let \( \xi \) be a random variable with distribution function \( F = F\left( x\right) \) and\n\n\[ \varphi \left( t\right) = \mathrm{E}{e}^{it\xi } \]\n\nits characteristic function. Then \( \varphi \) has the following properties:\n\n(1) \( \left| {\varphi \left( t\right) }\right| \leq \varphi \left( 0\righ... | Proof. Properties (1) and (3) are evident. Property (2) follows from the inequality\n\n\[ \left| {\varphi \left( {t + h}\right) - \varphi \left( t\right) }\right| = \left| {\mathrm{E}{e}^{it\xi }\left( {{e}^{ih\xi } - 1}\right) }\right| \leq \mathrm{E}\left| {{e}^{ih\xi } - 1}\right| \]\n\nand the dominated convergence... | Yes |
For any pair of points \( a \) and \( b\left( {a < b}\right) \) at which \( F = F\left( x\right) \) is continuous,\n\n\[ F\left( b\right) - F\left( a\right) = \mathop{\lim }\limits_{{c \rightarrow \infty }}\frac{1}{2\pi }{\int }_{-c}^{c}\frac{{e}^{-{ita}} - {e}^{-{itb}}}{it}\varphi \left( t\right) {dt}. \] | We have\n\n\[ {\Phi }_{c} \equiv \frac{1}{2\pi }{\int }_{-c}^{c}\frac{{e}^{-{ita}} - {e}^{-{itb}}}{it}\varphi \left( t\right) {dt} \]\n\n\[ = \frac{1}{2\pi }{\int }_{-c}^{c}\frac{{e}^{-{ita}} - {e}^{-{itb}}}{it}\left\lbrack {{\int }_{-\infty }^{\infty }{e}^{itx}{dF}\left( x\right) }}\right\rbrack {dt} \]\n\n\[ = \frac{... | Yes |
A necessary and sufficient condition for the components of the random vector \( \xi = {\left( {\xi }_{1},\ldots ,{\xi }_{n}\right) }^{ * } \) to be independent is that its characteristic function is the product of the characteristic functions of the components: | The necessity follows from Problem 1. To prove the sufficiency we let \( F\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be the distribution function of the vector \( \xi = {\left( {\xi }_{1},\ldots ,{\xi }_{n}\right) }^{ * } \) and \( {F}_{k}\left( x\right) \), the distribution functions of the \( {\xi }_{k},1 \leq k \leq... | Yes |
Theorem 5. Let \( {\varphi }_{\xi }\left( t\right) \) be the characteristic function of the random variable \( \xi \). (a) If \( \left| {{\varphi }_{\xi }\left( {t}_{0}\right) }\right| = 1 \) for some \( {t}_{0} \neq 0 \), then \( \xi \) is a lattice random variable concentrated at the points \( a + {nh}, h = {2\pi }/\... | Proof. (a) If \( \left| {{\varphi }_{\xi }\left( {t}_{0}\right) }\right| = 1,{t}_{0} \neq 0 \), there is a number \( a \) such that \( \varphi \left( {t}_{0}\right) = {e}^{i{t}_{0}a} \). Then\n\n\[ {e}^{i{t}_{0}a} = {\int }_{-\infty }^{\infty }{e}^{i{t}_{0}x}{dF}\left( x\right) \Rightarrow 1 = {\int }_{-\infty }^{\inft... | Yes |
Theorem 6. Let \( \xi = {\left( {\xi }_{1},\ldots ,{\xi }_{k}\right) }^{ * } \) be a random vector with \( \mathrm{E}{\left| {\xi }_{i}\right| }^{n} < \infty, i = \) \( 1,\ldots, k, n \geq 1 \) . Then for \( \nu = \left( {{\nu }_{1},\ldots ,{\nu }_{k}}\right) \) such that \( \left| \nu \right| \leq n \)\n\n\[ \n{m}_{\x... | Proof. Since\n\n\[ \n{\varphi }_{\xi }\left( t\right) = \exp \left( {\log {\varphi }_{\xi }\left( t\right) }\right) \n\]\n\nif we expand the function exp by Taylor's formula and use (39), we obtain\n\n\[ \n{\varphi }_{\xi }\left( t\right) = 1 + \mathop{\sum }\limits_{{q = 1}}^{n}\frac{1}{q!}{\left( \mathop{\sum }\limit... | Yes |
Corollary 1. The following formulas connect moments and semi-invariants:\n\n\[ \n{m}_{\xi }^{\left( \nu \right) } = \mathop{\sum }\limits_{\left\{ {r}_{1}{\lambda }^{\left( 1\right) } + \cdots + {r}_{x}{\lambda }^{\left( x\right) } = v\right\} }\frac{1}{{r}_{1}!\cdots {r}_{x}!}\frac{\nu !}{{\left( {\lambda }^{\left( 1\... | To establish (44) we suppose that among all the vectors \( {\lambda }^{\left( 1\right) },\ldots ,{\lambda }^{\left( q\right) } \) that occur in (40), there are \( {r}_{1} \) equal to \( {\lambda }^{\left( {i}_{1}\right) },\ldots ,{r}_{x} \) equal to \( {\lambda }^{\left( {i}_{x}\right) }({r}_{j} > 0 \) , \( \left. {{r}... | Yes |
Corollary 2. Let us consider the special case when \( \nu = \left( {1,\ldots ,1}\right) \) . In this case the moments \( {m}_{\xi }^{\left( \nu \right) } \equiv \mathbf{E}{\xi }_{1}\cdots {\xi }_{k} \), and the corresponding semi-invariants are called simple. | In accordance with the definition given in Subsection 3 of Sect. 1, Chap. 1, a decomposition of a set \( I \) is an unordered collection of disjoint nonempty sets \( {I}_{p} \) such that \( \mathop{\sum }\limits_{p}{I}_{p} = I \) . In terms of these definitions, we have the formulas \[ {m}_{\xi }\left( I\right) = \math... | Yes |
Let \( \xi \sim \mathcal{N}\left( {m,{\sigma }^{2}}\right) \). Since, by (9), | \[ \log {\varphi }_{\xi }\left( t\right) = {itm} - \frac{{t}^{2}{\sigma }^{2}}{2} \] we have \( {s}_{1} = m,{s}_{2} = {\sigma }^{2} \) by (39), and all the semi-invariants, from the third on, are zero: \( {s}_{n} = 0, n \geq 3 \) . | Yes |
If \( \xi \) is a Poisson random variable with parameter \( \lambda > 0 \), then by (11) | \[ \log {\varphi }_{\xi }\left( t\right) = \lambda \left( {{e}^{it} - 1}\right) \] It follows that \[ {s}_{n} = \lambda \] for all \( n \geq 1 \) . | Yes |
Example 7. Let \( \xi = {\left( {\xi }_{1},\ldots ,{\xi }_{n}\right) }^{ * } \) be a random vector. Then\n\n\[ \n{m}_{\xi }\left( 1\right) = {s}_{\xi }\left( 1\right) \]\n\n\[ \n{m}_{\xi }\left( {1,2}\right) = {s}_{\xi }\left( {1,2}\right) + {s}_{\xi }\left( 1\right) {s}_{\xi }\left( 2\right) \]\n\n\[ \n{m}_{\xi }\left... | These formulas show that the simple moments can be expressed in terms of the simple semi-invariants in a very symmetric way. If we put \( {\xi }_{1} \equiv {\xi }_{2} \equiv \cdots \equiv {\xi }_{k} \), we then, of course, obtain (48).\n\nThe group-theoretical origin of the coefficients in (48) becomes clear from (51).... | Yes |
Theorem 7. Let \( F = F\left( x\right) \) be a distribution function and \( {\mu }_{n} = {\int }_{-\infty }^{\infty }{\left| x\right| }^{n}{dF}\left( x\right) \) . If\n\n\[ \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{{\mu }_{n}^{1/n}}{n} < \infty \]\n\nthe moments \( {\left\{ {m}_{n}\right\} }_{n \geq 1} \)... | Proof. It follows from (56) and conclusion (7) of Theorem 1 that there is a \( {t}_{0} > 0 \) such that, for all \( \left| t\right| \leq {t}_{0} \), the characteristic function\n\n\[ \varphi \left( t\right) = {\int }_{-\infty }^{\infty }{e}^{itx}{dF}\left( x\right) \]\n\ncan be represented in the form\n\n\[ \varphi \le... | Yes |
Corollary 4. A sufficient condition for the moment problem to have a unique solution is that\n\n\[ \mathop{\limsup }\limits_{{n \rightarrow \infty }}\frac{{\left( {m}_{2n}\right) }^{1/{2n}}}{2n} < \infty . \] | For the proof it is enough to observe that the odd moments can be estimated in terms of the even ones, and then use (56). | No |
Theorem 2 (Theorem on Normal Correlation). For a Gaussian vector \( {\left( {\theta }^{ * },{\xi }^{ * }\right) }^{ * } \) , the optimal estimator \( \mathrm{E}\left( {\theta \mid \xi }\right) \) of \( \theta \) in terms of \( \xi \), and its error matrix\n\n\[ \n\Delta = \mathrm{E}\left\lbrack {\theta - \mathrm{E}\lef... | Proof. Form the vector\n\n\[ \n\eta = \left( {\theta - {m}_{\theta }}\right) - {\mathbb{R}}_{\theta \xi }{\mathbb{R}}_{\xi \xi }^{-1}\left( {\xi - {m}_{\xi }}\right)\n\]\n\n(21)\n\nWe can verify at once that \( \mathrm{E}\eta {\left( \xi - {m}_{\xi }\right) }^{ * } = 0 \), i.e., \( \eta \) is not correlated with \( \le... | Yes |
If \( \mathfrak{A} = \lbrack 0,\infty ) \) and\n\n\[ r\left( {s, t}\right) = \min \left( {s, t}\right) \]\n\nthe Gaussian process \( B = {\left( {B}_{t}\right) }_{t \geq 0} \) with this covariance function (see Problem 2) and \( {B}_{0} \equiv 0 \) is a Brownian motion or Wiener process. | Observe that this process has independent increments; that is, for arbitrary \( {t}_{1} < \) \( {t}_{2} < \cdots < {t}_{n} \) the random variables\n\n\[ {B}_{{t}_{2}} - {B}_{{t}_{1}},\ldots ,{B}_{{t}_{n}} - {B}_{{t}_{n - 1}} \]\n\nare independent. In fact, because the process is Gaussian it is enough to verify only tha... | Yes |
Theorem 3. Let \( B = {\left( {B}_{t}\right) }_{t \geq 0} \) be the standard Brownian motion. Then, with probability one,\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\sum }\limits_{{k = 1}}^{{{2}^{n}T}}{\left\lbrack {B}_{k{2}^{-n}} - {B}_{\left( {k - 1}\right) {2}^{-n}}\right\rbrack }^{2} = T \]\n\nfor... | Proof. Without loss of generality we can take \( T = 1 \) . Let\n\n\[ {A}_{n}^{\varepsilon } = \left\{ {\omega : \left| {\mathop{\sum }\limits_{{k = 1}}^{{2}^{n}}{\left( {B}_{k{2}^{-n}} - {B}_{\left( {k - 1}\right) {2}^{-n}}\right) }^{2} - 1}\right| \geq \varepsilon }\right\} .\n\nSince the random variables \( {B}_{k{2... | Yes |
Theorem 1. The following statements are equivalent.\n\n(I) \( {\mathrm{P}}_{n}\overset{w}{ \rightarrow }\mathrm{P} \) ,\n\n(II) \( \limsup {\mathrm{P}}_{n}\left( A\right) \leq \mathrm{P}\left( A\right) ,\;A \) closed,\n\n(III) \( \liminf {\mathrm{P}}_{n}\left( A\right) \geq \mathrm{P}\left( A\right) ,\; \) A open,\n\n(... | Proof. (I) \( \Rightarrow \) (II). Let \( A \) be closed and\n\n\[ \n{f}_{A}^{\varepsilon }\left( x\right) = {\left\lbrack 1 - \frac{\rho \left( {x, A}\right) }{\varepsilon }\right\rbrack }^{ + },\;\varepsilon > 0, \n\]\n\nwhere\n\n\[ \n\rho \left( {x, A}\right) = \inf \{ \rho \left( {x, y}\right) : y \in A\} ,\;{\left... | Yes |
Theorem 2. The following conditions are equivalent:\n\n(1) \( {P}_{n}\overset{w}{ \rightarrow }P \) ,\n\n(2) \( {P}_{n} \Rightarrow P \) ,\n\n(3) \( {F}_{n}\overset{w}{ \rightarrow }F \) ,\n\n(4) \( {F}_{n} \Rightarrow F \) . | Proof. Since \( \left( 2\right) \Leftrightarrow \left( 1\right) \Leftrightarrow \left( 3\right) \), it is enough to show that \( \left( 2\right) \Leftrightarrow \left( 4\right) \) .\n\nIf \( {P}_{n} \Rightarrow P \), then in particular\n\n\[ \left. {{P}_{n}( - \infty, x}\right\rbrack \rightarrow P( - \infty, x\rbrack \... | Yes |
Theorem 1 (Prokhorov’s Theorem). Let \( \mathcal{P} = \left\{ {{\mathrm{P}}_{\alpha };\alpha \in \mathfrak{A}}\right\} \) be a family of probability measures defined on a complete separable metric space \( \left( {E,\mathcal{E},\rho }\right) \) . Then \( \mathcal{P} \) is relatively compact if and only if it is tight. | Proof. We shall give the proof only when the space is the real line. (The proof can be carried over (see [9], [76]), almost unchanged, to arbitrary Euclidean spaces \( {R}^{n}, n \geq 2 \) . Then the theorem can be extended successively to \( {R}^{\infty } \), to \( \sigma \) -compact spaces; and finally to general com... | No |
Theorem 2 (Helly’s Theorem). The class \( \mathcal{I} = \{ G\} \) of generalized distribution functions is sequentially compact, i.e., for every sequence \( \left\{ {G}_{n}\right\} \) of functions from \( \mathcal{I} \) we can find a function \( G \in \mathcal{I} \) and a subsequence \( \left\{ {n}_{k}\right\} \subsete... | Proof. Let \( T = \left\{ {{x}_{1},{x}_{2},\ldots }\right\} \) be a countable dense subset of \( R \) . Since the sequence of numbers \( \left\{ {{G}_{n}\left( {x}_{1}\right) }\right\} \) is bounded, there is a subsequence \( {N}_{1} = \left\{ {{n}_{1}^{\left( 1\right) },{n}_{2}^{\left( 1\right) },\ldots }\right\} \) s... | Yes |
Theorem 1 (Continuity Theorem). Let \( \\left\\{ {F}_{n}\\right\\} \) be a sequence of distribution functions \( {F}_{n} = {F}_{n}\\left( x\\right), x \\in R \\), and let \( \\left\\{ {\\varphi }_{n}\\right\\} \) be the corresponding sequence of characteristic functions,\n\n\[ \n{\\varphi }_{n}\\left( t\\right) = {\\in... | The proof of conclusion (1) is an immediate consequence of the definition of weak convergence, applied to the functions \( \\operatorname{Re}{e}^{itX} \) and \( \\operatorname{Im}{e}^{itx} \) . | No |
Lemma 1. Let \( \\left\\{ {\\mathrm{P}}_{n}\\right\\} \) be a tight family of probability measures. Suppose that every weakly convergent subsequence \( \\left\\{ {\\mathrm{P}}_{{n}^{\\prime }}\\right\\} \) of \( \\left\\{ {\\mathrm{P}}_{n}\\right\\} \) converges to the same probability measure \( \\mathrm{P} \) . Then ... | Proof. Suppose that \( {\\mathrm{P}}_{n} \\nrightarrow \\mathrm{P} \) . Then there is a bounded continuous function \( f = \) \( f\\left( x\\right) \) such that\n\n\[{\\int }_{R}f\\left( x\\right) {\\mathrm{P}}_{n}\\left( {dx}\\right) \\nrightarrow {\\int }_{R}f\\left( x\\right) \\mathrm{P}\\left( {dx}\\right) .\]\n\nI... | Yes |
Lemma 2. Let \( \\left\\{ {\\mathrm{P}}_{n}\\right\\} \) be a tight family of probability measures on \( \\left\\{ {R},\\mathcal{B}\\left( R\\right) \\right\\} \) . A necessary and sufficient condition for the sequence \( \\left\\{ {\\mathrm{P}}_{n}\\right\\} \) to converge weakly to a probability measure is that for e... | Proof. If \( \\left\\{ {\\mathrm{P}}_{n}\\right\\} \) is tight, by Prohorov’s theorem there is a subsequence \( \\left\\{ {\\mathrm{P}}_{{n}^{\\prime }}\\right\\} \) and a probability measure \( \\mathrm{P} \) such that \( {\\mathrm{P}}_{{n}^{\\prime }}\\overset{w}{ \\rightarrow }\\mathrm{P} \) . Suppose that the whole... | Yes |
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