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Theorem 1. Suppose that\n\n(a) two functions \( p \) and \( q \) are analytic at a point \( {z}_{0} \) ;\n\n(b) \( p\left( {z}_{0}\right) \neq 0 \) and \( q \) has a zero of order \( m \) at \( {z}_{0} \) .\n\nThen the quotient \( p\left( z\right) /q\left( z\right) \) has a pole of order \( m \) at \( {z}_{0} \) . | The proof is easy. Let \( p \) and \( q \) be as in the statement of the theorem. Since \( q \) has a zero of order \( m \) at \( {z}_{0} \), we know from Theorem 2 in Sec. 75 that there is a deleted neighborhood of \( {z}_{0} \) throughout which \( q\left( z\right) \neq 0 \) ; and so \( {z}_{0} \) is an isolated singu... | Yes |
Theorem 2. Let two functions \( p \) and \( q \) be analytic at a point \( {z}_{0} \) . If\n\n\[ p\left( {z}_{0}\right) \neq 0,\;q\left( {z}_{0}\right) = 0,\;\text{ and }\;{q}^{\prime }\left( {z}_{0}\right) \neq 0, \]\n\nthen \( {z}_{0} \) is a simple pole of the quotient \( p\left( z\right) /q\left( z\right) \) and\n\... | To show this, we assume that \( p \) and \( q \) are as stated and observe that because of the conditions on \( q \), the point \( {z}_{0} \) is a zero of order \( m = 1 \) of that function. According to Theorem 1 in Sec. 75, then,\n\n(3)\n\n\[ q\left( z\right) = \left( {z - {z}_{0}}\right) g\left( z\right) \]\n\nwhere... | Yes |
Theorem 1. If \( {z}_{0} \) is a pole of a function \( f \), then\n\n(1)\n\n\[ \mathop{\lim }\limits_{{z \rightarrow {z}_{0}}}f\left( z\right) = \infty \] | To verify limit (1), we assume that \( f \) has a pole of order \( m \) at \( {z}_{0} \) and use the theorem in Sec. 73. It tells us that\n\n\[ f\left( z\right) = \frac{\phi \left( z\right) }{{\left( z - {z}_{0}\right) }^{m}} \]\n\nwhere \( \phi \left( z\right) \) is analytic and nonzero at \( {z}_{0} \) . Since\n\n\[ ... | Yes |
Theorem 2. If \( {z}_{0} \) is a removable singular point of a function \( f \), then \( f \) is analytic and bounded in some deleted neighborhood \( 0 < \left| {z - {z}_{0}}\right| < \varepsilon \) of \( {z}_{0} \) . | The proof is easy and is based on the fact that the function \( f \) here is analytic in a disk \( \left| {z - {z}_{0}}\right| < {R}_{2} \) when \( f\left( {z}_{0}\right) \) is properly defined; \( f \) is then continuous in any closed disk \( \left| {z - {z}_{0}}\right| \leq \varepsilon \) where \( \varepsilon < {R}_{... | Yes |
Theorem 3. Suppose that \( {z}_{0} \) is an essential singularity of a function \( f \), and let \( {w}_{0} \) be any complex number. Then, for any positive number \( \varepsilon \), the inequality\n\n(4)\n\n\[ \left| {f\left( z\right) - {w}_{0}}\right| < \varepsilon \]\n\nis satisfied at some point \( z \) in each del... | The proof is by contradiction. Since \( {z}_{0} \) is an isolated singularity of \( f \), there is a deleted neighborhood \( 0 < \left| {z - {z}_{0}}\right| < \delta \) throughout which \( f \) is analytic; and we\n\nassume that condition (4) is not satisfied for any point \( z \) there. Thus \( \left| {f\left( z\right... | Yes |
\[ P\left( {A \cup B}\right) = P\left( A\right) + P\left( B\right) - P\left( {AB}\right) . \] | Thus, the probability that the outcome of the experiment is either in \( A \) or in \( B \) equals the probability that it is in \( A \), plus the probability that it is in \( B \), minus the probability that it is in both \( A \) and \( B \) . | Yes |
Proposition 1.3.1 For random variables \( {X}_{1},\ldots ,{X}_{k} \) ,\n\n\[ E\left\lbrack {\mathop{\sum }\limits_{{j = 1}}^{k}{X}_{j}}\right\rbrack = \mathop{\sum }\limits_{{j = 1}}^{k}E\left\lbrack {X}_{j}\right\rbrack \] | Example 1.3d Consider \( n \) independent trials, each of which is a success with probability \( p \) . The random variable \( X \), equal to the total number of successes that occur, is called a binomial random variable with parameters \( n \) and \( p \) . We can determine its expectation by using the representation\... | Yes |
Proposition 1.3.2 If \( {X}_{1},\ldots ,{X}_{k} \) are independent random variables, then\n\n\[ \operatorname{Var}\left( {\mathop{\sum }\limits_{{j = 1}}^{k}{X}_{j}}\right) = \mathop{\sum }\limits_{{j = 1}}^{k}\operatorname{Var}\left( {X}_{j}\right) \] | Example 1.3g Find the variance of \( X \), a binomial random variable with parameters \( n \) and \( p \) .\n\nSolution. Recalling that \( X \) represents the number of successes in \( n \) independent trials (each of which is a success with probability \( p \) ), we can represent it as\n\n\[ X = \mathop{\sum }\limits_... | Yes |
Find the yield curve and the present value function if\n\n\\[ r\\left( s\\right) = \\frac{1}{1 + s}{r}_{1} + \\frac{s}{1 + s}{r}_{2} \\] | Solution. Rewriting \\( r\\left( s\\right) \\) as\n\n\\[ r\\left( s\\right) = {r}_{2} + \\frac{{r}_{1} - {r}_{2}}{1 + s},\\;s \\geq 0, \\]\n\nshows that the yield curve is given by\n\n\\[ \\ddot{r}\\left( t\\right) = \\frac{1}{t}{\\int }_{0}^{t}\\left( {{r}_{2} + \\frac{{r}_{1} - {r}_{2}}{1 + s}}\\right) {ds} \\]\n\n\\... | Yes |
Proposition 5.1.1 (The Law of One Price) Consider two investments, the first of which costs the fixed amount \( {C}_{1} \) and the second the fixed amount \( {C}_{2} \) . If the (present value) payoff from the first investment is always identical to that of the second investment, then either \( {C}_{1} = {C}_{2} \) or ... | The proof of the law of one price is immediate, because if their costs are unequal then an arbitrage is obtained by buying the cheaper investment and selling the more expensive one. | Yes |
Proposition 5.2.1 One should never exercise an American style call option before its expiration time \( t \) . | Proof. Suppose that the present price of the stock is \( S \), that you own an option to buy one share of the stock at a fixed price \( K \), and that the option expires after an additional time \( t \) . If you exercise the option at this moment, you will realize the amount \( S - K \) . However, consider what would t... | Yes |
Proposition 5.2.2 Let \( C \) be the price of a call option that enables its holder to buy one share of a stock at an exercise price \( K \) at time \( t \) ; also, let \( P \) be the price of a European put option that enables its holder to sell one share of the stock for the amount \( K \) at time \( t \) . Let \( S ... | Proof. If\n\n\[ S + P - C < K{e}^{-{rt}} \]\nthen we can effect a sure win by initially buying one share of the stock, buying one put option, and selling one call option. This initial payout of \( S + P - C \) is borrowed from a bank to be repaid at time \( t \) . Let us now consider the value of our holdings at time \... | Yes |
Proposition 5.2.3 (The Generalized Law of One Price) Consider two investments, the first of which costs the fixed amount \( {C}_{1} \) and the second the fixed amount \( {C}_{2} \) . If \( {C}_{1} < {C}_{2} \) and the (present value) payoff from the first investment is always at least as large as that from the second i... | The arbitrage is clearly obtained by simultaneously buying investment 1 and selling investment 2. | No |
Proposition 5.2.4 Let \( C\left( {K, t}\right) \) be the cost of a call option on a specified security that has strike price \( K \) and expiration time \( t \) .\n\n(a) For fixed expiration time \( t, C\left( {K, t}\right) \) is a convex and nonincreasing function of \( K \) . | Proof. If \( S\left( t\right) \) denotes the price of the security at time \( t \), then the payoff at time \( t \) from a \( \left( {K, t}\right) \) call option is\n\n\[ \text{ payoff of option } = \left\{ \begin{array}{ll} S\left( t\right) - K & \text{ if }S\left( t\right) \geq K, \\ 0 & \text{ if }S\left( t\right) <... | No |
Theorem 6.1.1 (The Arbitrage Theorem) Exactly one of the following is true: Either\n\n(a) there is a probability vector \( \\mathbf{p} = \\left( {{p}_{1},{p}_{2},\\ldots ,{p}_{m}}\\right) \) for which\n\n\[ \n\\mathop{\\sum }\\limits_{{j = 1}}^{m}{p}_{j}{r}_{i}\\left( j\\right) = 0\\;\\text{ for all }i = 1,\\ldots, n, ... | Proof. See Section 6.3. | No |
Proposition 6.3.2 (Arbitrage Theorem) Exactly one of the following is true: Either\n\n(i) there exists a probability vector \( \mathbf{p} = \left( {{p}_{1},\ldots ,{p}_{m}}\right) \) for which\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{m}{p}_{j}{r}_{i}\left( j\right) = 0\;\text{ for all }i = 1,\ldots, n \]\n\nor\n\n(ii) th... | Proof. Let \( {x}_{n + 1} \) denote an amount that the gambler can be sure of winning, and consider the problem of maximizing this amount. If the gambler uses the betting strategy \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) then she will win \( \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}{r}_{i}\left( j\right) \) if th... | Yes |
Lemma 7.5.1 Using the representations (7.3) and (7.4),\n\n\[ I = \left\{ \begin{array}{ll} 1 & \text{ if }Z > \sigma \sqrt{t} - \omega \\ 0 & \text{ otherwise } \end{array}\right. \]\n\nwhere\n\n\[ \omega = \frac{{rt} + {\sigma }^{2}t/2 - \log \left( {K/s}\right) }{\sigma \sqrt{t}}. \] | Proof.\n\n\[ S\left( t\right) > K \Leftrightarrow \exp \left\{ {\left( {r - {\sigma }^{2}/2}\right) t + \sigma \sqrt{t}Z}\right\} > K/s \]\n\n\[ \Leftrightarrow Z > \frac{\log \left( {K/s}\right) - \left( {r - {\sigma }^{2}/2}\right) t}{\sigma \sqrt{t}} \]\n\n\[ \Leftrightarrow Z > \sigma \sqrt{t} - \omega . \] | Yes |
\[ E\left\lbrack I\right\rbrack = P\{ S\left( t\right) > K\} = \Phi \left( {\omega - \sigma \sqrt{t}}\right) , \] where \( \Phi \) is the standard normal distribution function. | Proof. It follows from its definition that \[ E\left\lbrack I\right\rbrack = P\{ S\left( t\right) > K\} \] \[ = P\{ Z > \sigma \sqrt{t} - \omega \} \;\text{ (from Lemma 7.5.1) } \] \[ = P\{ Z < \omega - \sigma \sqrt{t}\} \] \[ = \Phi \left( {\omega - \sigma \sqrt{t}}\right) \] | Yes |
Lemma 7.5.3\n\n\\[ \n{e}^{-{rt}}E\\left\\lbrack {{IS}\\left( t\\right) }\\right\\rbrack = {s\\Phi }\\left( \\omega \\right) \n\\] | Proof. With \\( c = \\sigma \\sqrt{t} - \\omega \\), it follows from the representation (7.3) and Lemma 7.5.1 that\n\n\\[ \nE\\left\\lbrack {{IS}\\left( t\\right) }\\right\\rbrack = {\\int }_{c}^{\\infty }s\\exp \\left\\{ {\\left( {r - {\\sigma }^{2}/2}\\right) t + \\sigma \\sqrt{t}x}\\right\\} \\frac{1}{\\sqrt{2\\pi }... | Yes |
Theorem 7.5.1 (The Black-Scholes Pricing Formula)\n\n\[ C\left( {s, t, K,\sigma, r}\right) = {s\Phi }\left( \omega \right) - K{e}^{-{rt}}\Phi \left( {\omega - \sigma \sqrt{t}}\right) . \] | Proof.\n\n\[ C\left( {s, t, K,\sigma, r}\right) = {e}^{-{rt}}E\left\lbrack {\left( S\left( t\right) - K\right) }^{ + }\right\rbrack \]\n\n\[ = {e}^{-{rt}}E\left\lbrack {I\left( {S\left( t\right) - K}\right) }\right\rbrack \]\n\n\[ = {e}^{-{rt}}E\left\lbrack {I(S\left( t\right) \rbrack - K{e}^{-{rt}}E\left\lbrack I\righ... | Yes |
Proposition 7.5.1\n\n\[ \n\frac{\partial C}{\partial K} = - {e}^{-{rt}}\Phi \left( {\omega - \sigma \sqrt{t}}\right) \n\] | Proof. Because \( S\left( t\right) \) does not depend on \( K \) ,\n\n\[ \n\frac{\partial }{\partial K}{e}^{-{rt}}\left( {S\left( t\right) - K}\right) = - {e}^{-{rt}} \n\]\n\nUsing Equation (7.5), this gives\n\n\[ \n\frac{\partial C}{\partial K} = E\left\lbrack {-I{e}^{-{rt}}}\right\rbrack \n\]\n\n\[ \n= - {e}^{-{rt}}E... | Yes |
\[ \frac{\partial C}{\partial s} = \Phi \left( \omega \right) \] | Proof. Using the representation of Equation (7.3), we see that\n\n\[\frac{\partial }{\partial s}{e}^{-{rt}}\left( {S\left( t\right) - K}\right) = {e}^{-{rt}}\frac{\partial S\left( t\right) }{\partial s} = \frac{S\left( t\right) }{s}{e}^{-{rt}}.\]\n\nHence, by Equation (7.5),\n\n\[\frac{\partial C}{\partial s} = \frac{{... | Yes |
\[ \frac{\partial C}{\partial r} = {Kt}{e}^{-{rt}}\Phi \left( {\omega - \sigma \sqrt{t}}\right) \] | \[ \frac{\partial }{\partial r}\left\lbrack {{e}^{-{rt}}\left( {S\left( t\right) - K}\right) }\right\rbrack = - t{e}^{-{rt}}\left( {S\left( t\right) - K}\right) + {e}^{-{rt}}\frac{\partial S\left( t\right) }{\partial r} \] \[ = - t{e}^{-{rt}}\left( {S\left( t\right) - K}\right) + {e}^{-{rt}}{tS}\left( t\right) \;\text{... | Yes |
Lemma 7.5.4 With \( S\left( t\right) \) as given by Equation (7.3), \[ {e}^{-{rt}}E\left\lbrack {{IS}\left( t\right) Z}\right\rbrack = s\left( {{\Phi }^{\prime }\left( \omega \right) + \sigma \sqrt{t}\Phi \left( \omega \right) }\right) . \] | Proof. With \( c = \sigma \sqrt{t} - \omega \), it follows from Lemma 7.5.1 that \( E\left\lbrack {{IZS}\left( t\right) }\right\rbrack \) \[ = {\int }_{c}^{\infty }{xs}\exp \left\{ {\left( {r - {\sigma }^{2}/2}\right) t + \sigma \sqrt{t}x}\right\} \frac{1}{\sqrt{2\pi }}{e}^{-{x}^{2}/2}{dx} \] \[ = \frac{1}{\sqrt{2\pi }... | Yes |
\[ \frac{\partial C}{\partial \sigma } = s\sqrt{t}{\Phi }^{\prime }\left( \omega \right) \] | Proof. Equation (7.3) yields that \[ \frac{\partial }{\partial \sigma }\left\lbrack {{e}^{-{rt}}\left( {S\left( t\right) - K}\right) }\right\rbrack = {e}^{-{rt}}S\left( t\right) \left( {-{t\sigma } + \sqrt{t}Z}\right) . \] Hence, by Equation (7.5), \[ \frac{\partial C}{\partial \sigma } = E\left\lbrack {{e}^{-{rt}}{IS}... | Yes |
Proposition 7.5.5\n\n\\[\\frac{\\partial C}{\\partial t} = \\frac{\\sigma }{2\\sqrt{t}}s{\\Phi }^{\\prime }\\left( \\omega \\right) + {Kr}{e}^{-{rt}}\\Phi \\left( {\\omega - \\sigma \\sqrt{t}}\\right) .\\] | Proof.\n\n\\[\\frac{\\partial }{\\partial t}\\left\\lbrack {{e}^{-{rt}}\\left( {S\\left( t\\right) - K}\\right) }\\right\\rbrack = {e}^{-{rt}}\\frac{\\partial S\\left( t\\right) }{\\partial t} - r{e}^{-{rt}}S\\left( t\\right) + {Kr}{e}^{-{rt}}\\]\n\n\\[ = {e}^{-{rt}}S\\left( t\\right) \\left( {r - \\frac{{\\sigma }^{2}... | Yes |
Corollary 7.5.1 \( C\left( {s, t, K,\sigma, r}\right) \) is\n\n(a) decreasing and convex in \( K \) ;\n\n(b) increasing and convex in \( s \) ;\n\n(c) increasing, but neither convex nor concave, in \( r,\sigma \), and \( t \) . | Proof. (a) From Proposition 7.5.1, we have \( \frac{\partial C}{\partial K} < 0 \), and\n\n\[ \frac{{\partial }^{2}C}{\partial {K}^{2}} = - {e}^{-{rt}}{\Phi }^{\prime }\left( {\omega - \sigma \sqrt{t}}\right) \frac{\partial \omega }{\partial K} \]\n\n\[ = {e}^{-{rt}}{\Phi }^{\prime }\left( {\omega - \sigma \sqrt{t}}\ri... | Yes |
Theorem 8.4.1 If the jumps have a lognormal distribution with mean parameter \( {\mu }_{0} \) and variance parameter \( {\sigma }_{0}^{2} \), then the no-arbitrage cost of a European call option having strike price \( K \) and expiration time \( t \) is as follows:\n\n\[ \text{ no-arbitrage cost } = \mathop{\sum }\limi... | Remark. Although Theorem 8.4.1 involves an infinite series, in most applications \( \lambda \) - the rate at which jumps occur - will be quite small and thus the sum will converge rapidly. | No |
Theorem 8.4.2 Assuming a general distribution for the size of a jump, the\n\n\\[ \n\\text{no-arbitrage option cost} = E\\left\\lbrack {C\\left( {{s}_{t}J\\left( t\\right), t, K,\\sigma, r}\\right) }\\right\\rbrack \n\\]\n\n\\[ \n\\geq C\\left( {s, t, K,\\sigma, r}\\right) \\text{.} \n\\]\n\nMoreover,\n\nno-arbitrage op... | where\n\n\\[{s}_{t} = s{e}^{{\\lambda t}\\left( {1 - E\\left\\lbrack J\\right\\rbrack }\\right) }\\]\n\nand \\[\\omega = \\frac{{rt} + {\\sigma }^{2}t/2 - \\log \\left( {K/s}\\right) }{\\sigma \\sqrt{t}}.\\] | Yes |
Theorem 4 (Euclid’s second theorem). The number of primes is infinite. | We shall prove this in \( §{2.1} \) . | No |
Theorem 7 (TCHEBYCHEF’s THEOREM). The order of magnitude of \( \pi \left( x\right) \) is \( x/\log x \) : | \[ \pi \left( x\right) \asymp \frac{x}{\log x} \] | Yes |
Theorem 43 (Pythagoras’ theorem). \( \sqrt{}2 \) is irrational. | (i) First proof. If \( \sqrt{}2 \) is rational, then the equation\n\n(4.3.1)\n\n\[ \n{a}^{2} = 2{b}^{2} \n\]\n\nis soluble in integers \( a, b \) with \( \left( {a, b}\right) = 1 \) . Hence \( b \mid {a}^{2} \) and therefore \( p \mid {a}^{2} \) for any prime factor \( p \) of \( b \) . It follows that \( p \mid a \) .... | Yes |
Theorem 50. If\n\n\[ \n a = \mathop{\prod }\limits_{p}{p}^{\alpha }\;\left( {\alpha \geq 0}\right) ,{}^{ \dagger } \n\]\n\nand\n\n\[ \n b = \mathop{\prod }\limits_{p}{p}^{\beta }\;\left( {\beta \geq 0}\right) \n\]\n\nthen\n\n\[ \n \left( {a, b}\right) = \mathop{\prod }\limits_{p}{p}^{\min \left( {\alpha ,\beta }\right)... | This theorem is an immediate consequence of Theorem 2 and the definition of \( \left( {a, b}\right) \) . | No |
Theorem 69. If\n\n(i) \( P\\left( a\\right) \) and \( P\\left( b\\right) \) imply \( P\\left( {a + b}\\right) \) and \( P\\left( {a - b}\\right) \), for every \( a \) and \( b \) (provided, in the second case, that \( b \\leq a \) ),\n\n(ii) \( r \) is the least positive integer for which \( P\\left( r\\right) \) is tr... | In the first place, \( \\left( a\\right) \) is obvious.\n\nTo prove \( \\left( b\\right) \) we observe that \( 0 < r \\leq q \), by the definition of \( r \) . Hence we can write\n\n\[ q = {kr} + s,\\;s = q - {kr}, \]\n\nwhere \( k \\geq 1 \) and \( 0 \\leq s < r \) . But \( P\\left( r\\right) \\rightarrow P\\left( {kr... | Yes |
Theorem 80 (Wilson's theorem):\n\n\\[ \n\\left( {p - 1}\\right) ! \\equiv - 1\\left( {\\;\\operatorname{mod}\\;p}\\right) \n\\] | Thus 11 | 3628801. The congruence\n\n\\[ \n\\left( {p - 1}\\right) ! + 1 \\equiv 0\\left( {\\;\\operatorname{mod}\\;{p}^{2}}\\right) \n\\]\n\nis true for\n\n\\[ \np = 5,\\;p = {13},\\;p = {563}, \n\\]\n\nbut for no other value of \\( p \\) less than 200000 . Apparently no general theorem concerning the congruence is kn... | No |
Theorem 129. \[ \mathop{\prod }\limits_{{t\left( m\right) }}t \equiv \pm 1\left( {\;\operatorname{mod}\;m}\right) \] where the negative sign is to be chosen when \( m \) is \( 4,{p}^{a} \), or \( 2{p}^{a} \), where \( p \) is an odd prime, and the positive sign in all other cases. | The case \( m = p \) is Wilson’s theorem. | No |
Theorem 138. The sequence of digits is formed by the primes in ascending order, is irrational. | (1) Let us assume that any arithmetical progression of the form\n\n\[ {k.10}^{s + 1} + 1\;\left( {k = 1,2,3,\ldots }\right) \]\n\ncontains primes. Then there are primes whose expressions in the decimal system contain an arbitrary number \( s \) of 0 ’s, followed by a 1 . Since the decimal contains such sequences, it do... | No |
Theorem 172. If\n\n\[ x = \frac{{P\zeta } + R}{{Q\zeta } + S} \]\n\nwhere \( \zeta > 1 \) and \( P, Q, R \), and \( S \) are integers such that\n\n\[ Q > S > 0,\;{PS} - {QR} = \pm 1, \]\n\nthen R/S and P/Q are two consecutive convergents to the simple continued fraction whose value is \( x \) . If \( R/S \) is the \( \... | We can develop \( P/Q \) in a simple continued fraction\n\n(10.10.1)\n\n\[ \frac{P}{Q} = \left\lbrack {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right\rbrack = \frac{{p}_{n}}{{q}_{n}} \]\n\nAfter Theorem 158, we may suppose \( n \) odd or even as we please. We shall choose \( n \) so that\n\n(10.10.2)\n\n\[ {PS} - {QR} = \pm 1 ... | Yes |
Theorem 184. If\n\n(10.15.5)\n\n\[ \left| {\frac{p}{q} - x}\right| < \frac{1}{2{q}^{2}} \]\n\nthen \( p/q \) is a convergent. | If (10.15.5) is true, then\n\n\[ \frac{p}{q} - x = \frac{\epsilon \theta }{{q}^{2}} \]\nwhere\n\[ \epsilon = \pm 1,\;0 < \theta < \frac{1}{2}. \]\n\nWe can express \( p/q \) as a finite continued fraction\n\n\[ \left\lbrack {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right\rbrack \]\n\nand since, by Theorem 158, we can make \( n... | Yes |
Theorem 198. If\n\n\\[ \n\\sum \\frac{1}{\\chi \\left( q\\right) } \n\\]\n\nis convergent, then the set of \\( \\xi \\) which satisfy (11.11.2) for an infinity of \\( q \\) is null. | We may suppose \\( 0 < \\xi < 1 \\) . We enclose every \\( p/q \\) for which \\( q \\geq N \\) in an interval\n\n\\[ \n\\frac{p}{q} - \\frac{1}{{q\\chi }\\left( q\\right) },\\;\\frac{p}{q} + \\frac{1}{{q\\chi }\\left( q\\right) } \n\\]\n\nThere are less than \\( q \\) values of \\( p \\) corresponding to a given \\( q ... | Yes |
Theorem 229. If \( {\xi }^{3} + {\eta }^{3} + {\zeta }^{3} = 0 \), then one of \( \xi ,\eta ,\zeta \) is divisible by \( \lambda \) . | Let us suppose the contrary. Then\n\n\[ 0 = {\xi }^{3} + {\eta }^{3} + {\zeta }^{3} \equiv \pm 1 \pm 1 \pm 1\left( {\;\operatorname{mod}\;{\lambda }^{4}}\right) \]\n\nand so \( \pm 1 \equiv 0 \) or \( \pm 3 \equiv 0 \), i.e. \( {\lambda }^{4}\left| {1\text{or}{\lambda }^{4}}\right| 3 \) . The first hypothesis is\n\nunt... | Yes |
Theorem 266. If\n\n\[ g\left( n\right) = \mathop{\sum }\limits_{{d \mid n}}f\left( d\right) \]\n\nthen\n\n\[ f\left( n\right) = \mathop{\sum }\limits_{{d \mid n}}\mu \left( \frac{n}{d}\right) g\left( d\right) = \mathop{\sum }\limits_{{d \mid n}}\mu \left( d\right) g\left( \frac{n}{d}\right) . \] | In fact\n\n\[ \mathop{\sum }\limits_{{d \mid n}}\mu \left( d\right) g\left( \frac{n}{d}\right) = \mathop{\sum }\limits_{{d \mid n}}\mu \left( d\right) \mathop{\sum }\limits_{{c \mid \frac{n}{d}}}f\left( c\right) = \mathop{\sum }\limits_{{{cd} \mid n}}\mu \left( d\right) f\left( c\right) \]\n\n\[ = \mathop{\sum }\limits... | Yes |
Theorem 268. If\n\n\\[ G\\left( x\\right) = \\mathop{\\sum }\\limits_{{n = 1}}^{\\left\\lbrack x\\right\\rbrack }F\\left( \\frac{x}{n}\\right) \\]\n\nfor all positive \\( x,{}^{ \\dagger } \\) then\n\n\\[ F\\left( x\\right) = \\mathop{\\sum }\\limits_{{n = 1}}^{\\left\\lbrack x\\right\\rbrack }\\mu \\left( n\\right) G\... | For\n\n\\[ \\mathop{\\sum }\\limits_{{n = 1}}^{\\left\\lbrack x\\right\\rbrack }\\mu \\left( n\\right) G\\left( \\frac{x}{n}\\right) = \\mathop{\\sum }\\limits_{{n = 1}}^{\\left\\lbrack x\\right\\rbrack }\\mu \\left( n\\right) \\mathop{\\sum }\\limits_{{m = 1}}^{\\left\\lbrack x/n\\right\\rbrack }F\\left( \\frac{x}{mn}... | Yes |
Theorem 399. If \( \theta \geq 2,{}^{ \dagger } \) then \( G\left( {3.2}^{\theta }\right) \geq {2}^{\theta + 2} \) . | This is a trivial corollary of Theorem 396, since \( G\left( {3.2}^{\theta }\right) \geq G\left( {2}^{\theta }\right) \geq \) \( {2}^{\theta + 2} \) . | Yes |
Theorem 419. If\n\n\\[ \alpha = \\mathop{\\sum }\\limits_{{m = 1}}^{\\infty }{p}_{m}{10}^{-{2}^{m}} = \\cdot {020300050000000070}\\ldots ,\\]\n\nwe have\n\n(22.3.6)\n\n\\[ {p}_{n} = \\left\\lbrack {{10}^{{2}^{n}}\\alpha }\\right\\rbrack - {10}^{{2}^{n - 1}}\\left\\lbrack {{10}^{{2}^{n - 1}}\\alpha }\\right\\rbrack .\\] | By (2.2.2),\n\n\\[ {p}_{m} < {2}^{{2}^{m}} = {4}^{{2}^{m - 1}} \\]\n\nand so the series for \\( \\alpha \\) is convergent. Again\n\n\\[ 0 < {10}^{{2}^{n}}\\mathop{\\sum }\\limits_{{m = n + 1}}^{\\infty }{p}_{m}{10}^{-{2}^{m}} < \\mathop{\\sum }\\limits_{{m = n + 1}}^{\\infty }{4}^{{2}^{m - 1}}{10}^{-{2}^{m - 1}} \\]\n\... | Yes |
Theorem 442. If\n\n\[ \n{\vartheta }_{1},{\vartheta }_{2},\ldots ,{\vartheta }_{k},1 \n\]\n\nare linearly independent, \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{k} \) are arbitrary, and \( N \) and \( \epsilon \) are positive, then there are integers\n\n\[ \n n > N,\;{p}_{1},{p}_{2},\ldots ,{p}_{k} \n\]\n\nsuch... | (1) Theorem 444 implies Theorem 442. We suppose, as we may, that every \( \vartheta \) lies in \( \left( {0,1}\right) \) and that \( \epsilon < 1 \) . We apply Theorem 444, with \( k + 1 \) for \( k, N + 1 \) for \( T \), and \( \frac{1}{2} \in \) for \( \epsilon \), to the systems\n\n\[ \n{\vartheta }_{1},{\vartheta }... | Yes |
Theorem 475. Let \( E \) be an elliptic curve given by an equation (25.1.3) having rational coefficients. Then the quotient group \( E\left( \mathbb{Q}\right) /{2E}\left( \mathbb{Q}\right) \) is finite, i.e. there is a finite set of points \( {Q}_{1},\ldots ,{Q}_{k} \in E\left( \mathbb{Q}\right) \) such that every \( Q... | The second part of the proof of Theorem 470 is a descent argument very much in the spirit of Fermat. Making a change of varibles of the form \( x = {u}^{2}{x}^{\prime } \) and \( y = {u}^{3}{y}^{\prime } \) for an appropriate rational number \( u \), we may assume that the equation (25.1.3) defining \( E \) has integer... | Yes |
Theorem 2 Let \( \left( {{p}_{1},\cdots ,{p}_{r}}\right) \) be a privileged basis of any irreducible component \( {V}^{\prime } \) of the associated variety \( {V}^{ * } \) . There is an algorithmic procedure which permits us to determine, for any polynomial \( h \) in \( R\left\lbrack {{u}_{1},\cdots ,{u}_{d},{x}_{1},... | The polynomials \( {h}_{{i}_{1}},\cdots ,{i}_{r} \) which are uniquely determined up to multipliers in \( R \) by the algorithmic procedure, will be called the remainder constituents of the polynomial \( h \) with respect to the privileged basis \( \left( {{p}_{1},\cdots ,{p}_{r}}\right) \) of \( {V}^{\prime } \), or b... | No |
Lemma 2. There is an algorithmic procedure which permits us to determine for any polynomial in some indeterminate \( y \) of the form\n\n\[ A = {A}_{0}{y}^{m} + {A}_{1}{y}^{m - 1} + \cdots + {A}_{m} \]\n\nwith each \( {A}_{i} \) in \( {p}_{r} \) and \( {A}_{0} \neq 0 \) in \( {K}_{r} \) expressions of the form\n\n\[ {H... | Proof. By lemma 1, we have for some integers \( {s}_{1},\cdots ,{s}_{r} \) \( \geq 0 \) an expression of the form\n\n\( {P}_{10}^{s}\cdots {P}_{r0}^{s}A = {A}^{\prime }{}_{0}{y}^{m} + {A}^{\prime }{}_{1}{y}^{m - 1} + \cdots + {A}^{\prime }{}_{m}{\;\operatorname{mod}\;\left( {{p}_{1},\cdots ,{p}_{r}}\right) } \)\n\nwith... | Yes |
Lemma 3. There is an algorithmic procedure which permits to factorize in \( {K}_{r} \) any polynomial in \( {p}_{r}\left\lbrack y\right\rbrack \)\n\n\[ A = {A}_{0}{y}^{m} + {A}_{1}{y}^{m - 1} + \cdots + {A}_{m} \]\n\nwith \( {A}_{i} \) in \( {p}_{r} \) and \( {A}_{0} \neq 0 \) in \( {K}_{r}, m \geq 2 \). More precisely... | Proof. The method of Hermann in [4] permits us to give, in an algorithmic manner, a factorization of \( A \) into irreducible ones in \( {K}_{r} \), so that after clearing of fractions we have an expression of the form\n\n\[ {DA} = {B}_{1}\cdots {B}_{t} + \sum {B}^{\prime }{}_{i}{p}_{i}, \]\n\nwith \( {B}_{i},{B}_{i}^{... | Yes |
To solve transient response of a 2-d frame structure. | The solution methods are: 1) central difference scheme. 3) Houbolt integration scheme. 4) Wilson ! É integration scheme. 5) Newmark integration scheme nodal dof: \( \{ {u1}\;{v1}\;{w1}\;\theta \;{x1}\;\theta \;{y1}\;\theta \;{z1}\;{u2}\;{v2}\;{w2}\;\theta \;{x2}\;\theta \;{y2}\;\theta \;{z2}\} \) | No |
Lemma 1.1.1 (Slepian 1962, Adler 1989) Let \( \{ X\left( t\right) ;t \in T\} \) and \( \{ Y\left( t\right) ;t \in T\} \) be centered Gaussian processes such that \( E{X}^{2}\left( t\right) = E{Y}^{2}\left( t\right) \) for all \( t \in T \) and \( {EY}\left( t\right) Y\left( s\right) \leq {EX}\left( t\right) X\left( s\r... | \[ P\left\{ {\mathop{\sup }\limits_{{t \in T}}X\left( t\right) \leq u}\right\} \geq P\left\{ {\mathop{\sup }\limits_{{t \in T}}Y\left( t\right) \leq u}\right\} . \] | Yes |
Lemma 1.1.2 (Révész 1982) Let \( k \) be an arbitrary positive number. Then for any \( \varepsilon > 0 \) there exists a \( {u}_{0} = {u}_{0}\left( {k,\varepsilon }\right) > 0 \) such that, for any \( u \geq {u}_{0} \) , we have\n\n\[ \left( {1 - \varepsilon }\right) \frac{ku}{\sqrt{2\pi }}{e}^{-{u}^{2}/2} \leq P\left\... | Proof The first inequality in (1.1.13) is well-known (see Qualls and Watanabe 1972). We need only to prove the last inequality. Let\n\n\[ {x}_{i} = i/{u}^{2},\;i = 1,2,\cdots ,{\left\lbrack {u}^{2}k\right\rbrack }^{\left( *\right) } \]\n\nbe a partition of the interval \( \left\lbrack {0, k}\right\rbrack \) and define ... | Yes |
Lemma 1.1.3 (Revesz 1982) For any \( \varepsilon > 0 \) there exists \( u = {u}_{0}\left( \varepsilon \right) > 0 \) and \( {T}_{0} = {T}_{0}\left( \varepsilon \right) > 0 \) such that\n\n\[ \exp \left\{ {-{25}\frac{Tu}{\sqrt{2\pi }} - {e}^{-{u}^{2}/2}}\right\} \leq P\left\{ {\mathop{\sup }\limits_{{0 \leq t \leq T}}\m... | Proof Let \( k = \left\lbrack T\right\rbrack \) . From Lemma 1.1.1 and Lemma 1.1.2, we have\n\n\[ P\left\{ {\mathop{\sup }\limits_{{0 \leq t \leq T}}\mathop{\sup }\limits_{{0 \leq s \leq 1}}\left( {W\left( {t + s}\right) - W\left( t\right) }\right) \leq u}\right\} \]\n\n\[ \geq P\left\{ {\mathop{\max }\limits_{{0 \leq ... | Yes |
Theorem 1.1.3 (Chen, Kong, Lin 1986)\n\n\[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{0 < t \leq T}}\left| {W\left( T\right) - W\left( {T - t}\right) }\right| /d\left( {T, t}\right) = 1\;\text{ a.s. } \] | Proof \( {1}^{ \circ } \) Using the law of the iterated logarithm, we have the left hand side of (1.1.25)\n\n\[ \geq \overline{\mathop{\lim }\limits_{{T \rightarrow \infty }}}\left| {W\left( T\right) }\right| /{\left( 2T\log \log T\right) }^{1/2} = 1\;\text{ a.s. } \] | No |
Theorem 1.1.4 Let \( {a}_{T},{b}_{T} \) and \( {c}_{T} \) be non-negative functions with \( {a}_{T} + {b}_{T} \geq {c}_{T} \rightarrow \infty \) as \( T \rightarrow \infty \) . If there exists a constant \( A > 0 \) such that for any \( T > 1 \) ,\n\n\[ \n{b}_{T} - {b}_{T - 1} \leq A{a}_{T},{a}_{T} + {b}_{T} \leq A\lef... | Proof \( \;{1}^{ \circ } \) We prove\n\n\[ \n\mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{0 \leq t}}\mathop{\sup }\limits_{{0 < s}}\mathop{\sup }\limits_{{0 \leq r \leq s}}\left| {W\left( {t + r}\right) - W\left( t\right) }\right| /d\left( {t + s \vee {c}_{T}, s}\right) \leq 1 \n\]\n\na.s. \( ... | Yes |
Theorem 1.2.1 (He, Chen 1989) Let \( 0 < {a}_{T} \leq T \) be a non-decreasing function of \( T \) and satisfy\n\n\[ \n\text{(iv)}\;\mathop{\lim }\limits_{{T \rightarrow \infty }}\left( {\log T/{a}_{T}}\right) /\log \log T = r\;0 \leq r \leq \infty \text{.}\n\]\n\nThen we have\n\n\[ \n\mathop{\lim }\limits_{{T \rightar... | Proof First, we prove that\n\n\[ \n\mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{{a}_{T} \leq t \leq T}}\mathop{\sup }\limits_{{t \leq s \leq T}}\left| {W\left( s\right) - W\left( {s - t}\right) }\right| /d\left( {T, t}\right) \geq {\alpha }_{r}\;\text{ a.s. }\n\]\n\n(1.2.3)\n\nIt is clear that... | Yes |
Lemma 1.2.1 (Strassen 1964) Define\n\n\[ \n{\eta }_{T}\left( x\right) = W\left( {Tx}\right) /{\left( 2T\log \log T\right) }^{1/2}\;0 \leq x \leq 1.\n\]\n\nThe sequence \( \left\{ {{\eta }_{T}\left( x\right) }\right\} \) is relatively compact in \( C\left\lbrack {0,1}\right\rbrack \) with probability one, and the set of... | The proof can be found in Csörgő and Revesz (1981, Theorem 1.3.2). | Yes |
Lemma 1.2.2 Suppose \( a < b \) and \( f\left( x\right) = {\alpha x} + \beta \) for \( x = a \) and \( x = b \) . Suppose also that \( f \) is absolutely continuous on \( \left\lbrack {a, b}\right\rbrack \) with the Radon-Nikodym derivative \( {f}^{\prime } \) . Then\n\[{\int }_{a}^{b}{\left( {f}^{\prime }\left( x\righ... | Proof Let \( \mu \) be Lebesgue measure, \( P = \mu /\left( {b - a}\right) \) . Then \( X = {f}^{\prime } \) is a random variable on the probability space \( \left( {\left\lbrack {a, b}\right\rbrack ,\sum, P}\right) \), where \( \sum \) is the collection of \( \mu \) -measurable subsets of \( \left\lbrack {a, b}\right\... | Yes |
Theorem 1.3.1 (Révész 1982) Let \( 0 < {a}_{T} \leq T \) be a function of \( T \) for which\n\n(i) \( {a}_{T} \) is non-decreasing,\n\n(ii) \( T/{a}_{T} \) is non-decreasing,\n\n(iii) \( \mathop{\lim }\limits_{{T \rightarrow \infty }}\left( {\log T/{a}_{T}}\right) /\log \log T = \infty \) ,\n\nand put\n\n\[ \n{a}_{1}\l... | Proof \( {1}^{ \circ } \) Proof of (1.3.1). Since\n\n\[ \n{Y}_{1}\left( T\right) = \min \left( {{Y}_{1}\left( T\right) ,{Y}_{2}\left( T\right) ,{Y}_{3}\left( T\right) ,{Y}_{4}\left( T\right) }\right) \n\]\n\n\[ \n\leq \max \left( {{Y}_{1}\left( T\right) ,{Y}_{2}\left( T\right) ,{Y}_{3}\left( T\right) ,{Y}_{4}\left( T\r... | Yes |
Corollary 1.3.1 Let \( {a}_{T} \) be as in Theorem 1.3.1 except that (iii) is replaced by the stronger condition\n\n\[ \n\\text{(iii ')}\\mathop{\\lim }\\limits_{{T \\rightarrow \\infty }}{\\left( \\log T/{a}_{T}\\right) }^{1/2}/\\log \\log T = \\infty \\text{.}\n\]\n\nThen we have\n\n\[ \n\\mathop{\\lim }\\limits_{{T ... | Remark 1.3.1 If Condition (iii ') does not hold true, then (1.3.8) does not hold as well. In fact, if\n\nthen\n\[ \n\\mathop{\\lim }\\limits_{{T \\rightarrow \\infty }}{\\left( \\log T/{a}_{T}\\right) }^{1/2}\\log \\log T = r > 0,\n\]\n\n(1.3.9)\n\n\[ \n0 = \\mathop{\\lim }\\limits_{\\overline{T \\rightarrow \\infty }}... | Yes |
Corollary 1.4.3 (Kong 1987) Suppose that \( {a}_{T} \) and \( {\lambda }_{T} \) satisfy conditions (i), (ii), (iii) in Theorem 1.4.2 and (iv) in Theorem 1.4.1. Then\n\n\[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{R \in {LT}}}{\lambda }_{T}\left| {W\left( R\right) }\right| = \mathop{\lim }\l... | Proof It is easy to see that \( {\delta }_{T} \) satisfies all the conditions in Theorem 1.4.1 now, i.e., \( {\delta }_{T} \) is non-increasing and\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}{\delta }_{{\theta }^{k}}/{\delta }_{{\theta }^{k + 1}} = \mathop{\lim }\limits_{{k \rightarrow \infty }}{\lambda }_{{\... | Yes |
Theorem 1.4.3 (Lu 1991a) Let \( {b}_{T} \geq {T}^{1/2} \) be a non-decreasing function of \( T \) . Denote \[ {L}_{T}^{ * }\left( t\right) = \left\{ {R : R \subset {D}_{T}\left( {b}_{T}\right) ,\lambda \left( R\right) = t}\right\} , \] \[ {L}_{T}\left( t\right) = \left\{ {R : R \subset {D}_{T}\left( {b}_{T}\right) ,\la... | Proof \( {1}^{ \circ } \) In order to prove \[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{0 < t \leq T}}\mathop{\sup }\limits_{{R \in {LT}\left( t\right) }}\left| {W\left( R\right) }\right| /{d}^{ * }\left( {T, t}\right) \leq 1\;\text{ a.s. } \] (1.4.34) we take a real number \( \theta > 1 \... | Yes |
Lemma 1.4.1 Suppose that \( {\gamma }_{T} \) satisfies the Conditions (ii ’) and (iii ’) of Theorem 1.4.3. Then we have\n\n\[ \mathop{\lim }\limits_{{a \rightarrow \infty }}\mathop{\sup }\limits_{{R \in {Lv}\left( {v - u}\right), a \leq v - u}}\left| {W\left( R\right) }\right| /{d}^{ * }\left( {v, v - u}\right) = 1\;\t... | The proof is quite similar to that of Corollary 1.1.1 and will not be presented here. | No |
Lemma 1.5.4 Let \( \left( {{A}_{n};n \geq 1}\right\} \) be a sequence of events. If\n\n(i) \( \mathop{\sum }\limits_{{n = 1}}^{\infty }P\left( {A}_{n}\right) = \infty \) ,\n\n(ii) \( \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\sum }\limits_{{1 \leq j < k \leq n}}\left\lbrack {P\left( {{A}_{j}{A}_{k}}\right)... | The proof of Lemma 1.5.4 can be found in Billingsley (1986). | No |
Theorem 1.5.2 Let \( \{ X\left( t\right) ;t \geq 0\} \) be as above, for which (1.5.1) is satisfied, then\n\n\[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{0 < t \leq T}}\left| {X\left( T\right) - X\left( {T - t}\right) }\right| /d\left( {T, t}\right) = 1\;\text{ a.s. } \]\n\n(1.5.12)\n\n\[ \... | The proof is analogous to that of Theorem 1.1.3 and Theorem 1.5.1. So we omit it here. | No |
Theorem 1.5.3 (Csaki et al. 1990) Let \( 0 < {a}_{T} \leq T \) be a function of T. If the Conditions (i) and (ii) of Theorem 1.1.1 are satisfied, in addition, for any \( 0 \leq a < b \leq c < d \) we have also\n\n\[ E\left( {\Gamma \left( b\right) - \Gamma \left( a\right) }\right) \left( {\Gamma \left( d\right) - \Gamm... | The proof of Theorem 1. 5. 3 is similar to that of Theorem 1. 2. 1 in Csörgő-Revesz (1981), here Lemma 1.5.4 is used with the help of Condition (1.5.16). The details are omitted. | No |
Theorem 2.1.1 Suppose that \( \\left\\{ {X}_{n}\\right\\} \) satisfies the condition\n\n\[ \n\\text{there exists a}{t}_{0} > 0\\text{such that}E{e}^{t{x}_{1}}\\text{is finite for}\\left| t\\right| < {t}_{0}.\\text{.}\n\]\n\n(2.1.1)\n\nSuppose that in addition to Conditions (i) and (ii), \( \\left\\{ {a}_{n}\\right\\} \... | \[ \n\\mathop{\\lim }\\limits_{{N \\rightarrow \\infty }}\\mathop{\\max }\\limits_{{1 \\leq k \\leq {a}_{N}}}{\\beta }_{N}\\left| {{S}_{N + k} - {S}_{N}}\\right| = 1\\;\\text{ a.s. }\n\]\n\n(2.1.4)\n\n\[ \n\\mathop{\\lim }\\limits_{{N \\rightarrow \\infty }}{\\beta }_{N}\\left| {{S}_{N + {a}_{N}} - {S}_{N}}\\right| = 1... | No |
Theorem 2.1.4 Let \( \left\{ {X}_{n}\right\} \) be a sequence of i.i.d. random variables with mean 0 and variance 1. Let \( \left\{ {a}_{n}\right\} \) be a non-decreasing sequence of integers satisfying Conditions (i), (ii) and (iii). Then we have\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\min }\limi... | They declare that this theorem can be proved by repeating the proof of the corresponding theorem for a Wiener process (Theorem 1.7.1 of Csörgő and Révész 1981). Unfortunately, this seems to be impossible from a careful investigation of that proof (check the proof from (1.7.4) to (1.7.5) on pages 49 -50 in that book). | No |
Theorem 2.2.2 Suppose that Conditions (i ) and (iii) are satisfied, and that \( 1 \leq {\varphi }_{n} \leq n \) and \( {\varphi }_{n}/\log n \rightarrow \infty \) as \( n \rightarrow \infty \) . Then\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\max }\limits_{{{\varphi }_{N} \leq k \leq N}}\left| {{S}_{... | In order to prove the theorem, with the help of Theorem 2.2.1, it is enough to show\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\max }\limits_{{{\varphi }_{N} \leq k \leq N}}\mathop{\max }\limits_{{1 \leq j \leq k}}\left| {{S}_{N} - {S}_{N - j}}\right| /g\left( {N, k}\right) \leq 1\;\text{ a.s. } \]\n\... | No |
Lemma 2.3.1 Let \( X \) be a random variable with \( {EX} = 0 \) . Let \( a > 0 \) and \( 0 \leq \alpha \leq 1 \) . Then for any \( t \geq 0 \), we have\n\n\[ E\exp \{ {tXI}\left( {X \leq a}\right) \} \leq \exp \left\{ {\frac{{t}^{2}}{2}E{X}^{2}\left| {{t}^{2 + \alpha }{e}^{2ta}E}\right| {X}^{2 + \alpha }}\right\} . \] | Proof Note that\n\n\[ E\exp \{ {tXI}\left( {X \leq a}\right) \}\]\n\n\[ = 1 + {tEXI}\left( {X \leq a}\right) + \frac{{t}^{2}}{2}E{X}^{2}I\left( {X \leq a}\right) + E\left\{ {\mathop{\sum }\limits_{{j = 3}}^{\infty }\frac{{t}^{j}{X}^{j}}{j!}I\left( {X \leq a}\right) }\right\}\]\n\n\[ \leq 1 + \frac{{t}^{2}}{2}E{X}^{2} +... | Yes |
Theorem 2.3.2 Suppose that \( \left\{ {X}_{n}\right\} \) satisfies the Conditions (i) and (ii ’) there exists a non-decreasing continuous function \( H\left( x\right), x \geq 0 \), satisfying\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }P\left\{ {H\left( \left| {X}_{n}\right| \right) > {bn}}\right\} < \infty \text{ f... | The proof of Theorem 2.3.2 is similar to that of Theorem 2.3.1 excetp that we use the following Lemma instead of Lemma 2.3.1.\n\nLemma 2.3.2 Let \( X \) | No |
Lemma 2.3.2 Let \( X \) be a random variable with \( {EX} = 0 \) . Let \( a > 0 \) , \( 0 < \alpha \leq 1 \) . Suppose that \( x/\log H\left( x\right) \left( {x > 0}\right) \) is non-decreasing. Then for \( 0 \leq {ta} \leq \left( {{\alpha }^{2}/{10}}\right) \log H\left( a\right) \), we have\n\n\[ E\exp \{ {tXI}\left( ... | The proof of this lemma is similar to that of Lemma 2. 3. 1 and is omitted here. | No |
Theorem 2.3.3 (Lin 1990a) Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a sequence of independent random variables. Suppose that the following conditions are satisfied :\n\n(i) \( \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\mathop{\\inf }\\limits_{{m > 0}}E{\\left( {X}_{m + 1} + \\cdots + {X}_{m + n}\... | Proof By Conditions (i) and (ii), it is easy to see that there exist \( 0 < {c}_{1} \\leq {c}_{2} < \\infty \) such that\n\n\[ \n{c}_{1}{a}_{N} \\leq {\\sigma }_{nN}^{2} \\leq {c}_{2}{a}_{N}\\text{ for large }N.\n\]\n\n(2.3.29)\n\nFirst, we prove\n\n\[ \n\\mathop{\\lim }\\limits_{{N \\rightarrow \\infty }}\\mathop{\\ma... | Yes |
Lemma 2.5.2 Let \( \\left\\{ {X}_{n}\\right\\} \) be a sequence of random variables satisfying the conditions of Theorem 2.5.1. Then\n\n\[ \n\\log P\\left\\{ {\\mathop{\\max }\\limits_{{1 \\leq k \\leq {a}_{N}}}\\left| {{S}_{n + k} - {S}_{n}}\\right| \\leq {x}_{nN}{\\sigma }_{n{a}_{N}}}\\right\\} \\sim - \\frac{{\\pi }... | The proof of Lemma 2.5.2 can be found in Shao (1989). | Yes |
Example 2.5.1 Let \( \left\{ {{X}_{n};n \geq 1}\right\} \) be a sequence of independent random variables with distribution :\n\n\[ P\left\{ {{X}_{n} = {n}^{1/2}\log \log n}\right\} = P\left\{ {{X}_{n} = - {n}^{1/2}\log \log n}\right\} = {\left( 2n{\left( \log \log n\right) }^{3}\right) }^{-1}, \]\n\n\[ P\left\{ {{X}_{n... | On the other hand, for any \( c > 0 \)\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }P\left\{ {\left| {X}_{n}\right| \geq c{\left( n\log \log n\right) }^{1/2}}\right\} = \infty . \]\n\nThus\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\left| {\mathop{\sum }\limits_{{i = 1}}^{N}{X}_{i}}\right| /{\left( 2N\log \... | Yes |
Theorem 2. 6. 1 (Komlos, Major, Tusnady 1975,1976) Let \( \\left\\{ {X}_{n}\\right. \) ; \( n \\geq 1\\} \) be a sequence of i.i.d. random variables with \( E{X}_{1} = 0, E{X}_{1}^{2} = 1 \) , \( E{\\left| {X}_{1}\\right| }^{p} < \\infty \) . Then, we can redefine \( \\left\\{ {{X}_{n};n \\geq 1}\\right\\} \) on a rich... | \[ \mathop{\\sum }\\limits_{{i = 1}}^{n}{X}_{i} - W\\left( n\\right) = o\\left( {n}^{1/p}\\right) \\;\\text{ a.s. } \] | Yes |
Theorem 2.6.3 Suppose that (2.6.2) is satisfied and that\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}E{\xi }_{i}^{2} \leq {B}_{n} \rightarrow \infty \text{ as }n \rightarrow \infty ,\] \n\nwhere \( \left\{ {B}_{n}\right\} \) is a sequence of positive numbers. Then, we have\n\n\[ \left| {\mathop{\sum }\limits_{{i = 1}}^{n}... | Proof Put \( {A}_{k} = \left\{ {n : {2}^{k} < {B}_{n} \leq {2}^{k + 1}}\right\} ,\mathcal{A} = \left\{ {k;{A}_{k} \neq \phi }\right\} \). Noting that \( {B}_{n} \rightarrow \infty \), we have that \( \mathcal{A} \) contains infinite numbers. Write \( \mathcal{A} = \) \( \left\{ {{k}_{1},{k}_{2},\cdots }\right\} ,{n}_{i... | Yes |
Theorem 2.6.7 (Shao 1989) Let \( \left\{ {{a}_{N};N \geq 1}\right\} \) and \( \left\{ {{b}_{N};N \geq 1}\right\} \) be sequences of non-negative integers and \( \{ H\left( n\right) ;n \geq 1\} \) and \( \left\{ {{X}_{n};n \geq 1}\right\} \) be as in Theorem 2.6.4 satisfying (2.6.8)- (2.6.10). Put\n\n\[ \n{\sigma }_{n, ... | Proof Put \( {\sigma }_{i}^{2} = \operatorname{Var}{X}_{i}I\left( {\left| {X}_{i}\right| \leq H\left( i\right) }\right) \) . By Theorems 2.6.4 and 2.6.6 and (2.6.32), we have\n\n\[ \n\mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\max }\limits_{{0 \leq n \leq {b}_{N}}}\mathop{\max }\limits_{{1 \leq j \leq {a}_{... | No |
Theorem 2.6.8 Let \( \{ H\left( n\right) ;n \geq 1\} \) be a non-decreasing sequence of positive numbers, \( \left\{ {{a}_{N};N \geq 1}\right\} \) and \( \left\{ {{b}_{N};N \geq 1}\right\} \) be sequences of nonnegative integers, and \( \left\{ {{X}_{n};n \geq 1}\right\} \) be a sequence of independent random variables... | \[ {b}_{N} - {b}_{N - 1} \leq A{a}_{N}, \] (2.6.39) \[ {b}_{N} + {a}_{N} \leq A\left( {{a}_{N - 1} + {b}_{N - 1}}\right) , \] \( \left( {2.6.40}\right) \) \[ {H}^{2}\left( n\right) /{n}^{\theta + \alpha }\text{is non-decreasing.} \] (2.6.41) \[ {a}_{N} \geq {C}_{1}\frac{{H}^{2}\left( {{b}_{N} + {a}_{N}}\right) }{{\left... | Yes |
Lemma 3.2.1 Suppose that there exists a constant \( A > 0 \) such that \[ \mathop{\sum }\limits_{\substack{{1 \leq i \leq N} \\ {{\lambda }_{i} > 1/h} }}{\gamma }_{i}/{\lambda }_{i} \leq {Ah}\mathop{\sum }\limits_{\substack{{1 \leq i \leq N} \\ {{\lambda }_{i} \leq 1/h} }}{\gamma }_{i}\;\text{ for }0 \leq h \leq {h}_{N... | Proof Let \( f\left( h\right) = {f}_{N}\left( h\right) = {\sigma }_{N}^{2}\left( h\right) /{h}^{x} \). For \( 0 < h < {h}_{N} \) we have \[ {f}^{\prime }\left( h\right) = {h}^{-\alpha - 1}\left( {-\alpha \mathop{\sum }\limits_{{i = 1}}^{N}\frac{{\gamma }_{i}}{{\lambda }_{i}}\left( {1 - {e}^{-{\lambda }_{i}h}}\right) + ... | Yes |
Lemma 3.2.4 Let \( \\left\\{ {T}_{N}\\right\\} ,\\left\\{ {h}_{N}\\right\\} ,\\left\\{ {h}_{N}^{\\prime }\\right\\} \) and \( \\left\\{ {h}_{N}^{\\prime }\\right\\} \) be sequences of positive numbers with \( {h}_{N} \\leq {T}_{N} \) and \( {h}_{N}^{\\prime } \\leq {h}_{N} \\leq {h}_{N}^{\\prime \\prime } \) . Suppose ... | Proof Let \( r = r\\left( \\varepsilon \\right) \) be a positive number specified later on. Put \( {r}_{1} = {h}_{N}^{\\prime }/ \) \( {2}^{{2}^{r}} \) and \( {t}_{r} = \\left\\lbrack {t/{r}_{1}}\\right\\rbrack {r}_{1} \) . We can also write inequality (3.2.3). By Lemma 3.2.1 (noting that \( {h}_{N} \\leq {h}_{n} \) fo... | No |
Lemma 3.2.5 For any \( \varepsilon > 0 \), there exist constants \( C = C\left( \varepsilon \right) > 0 \) and \( v\left( \varepsilon \right) > 0 \) such that the inequality\n\n\[ P\left\{ {\mathop{\max }\limits_{{1 \leq n \leq N}}\mathop{\sup }\limits_{{\left| t\right| \leq {T}_{N}}}\left| {X\left( {t, n}\right) }\rig... | Proof We prove that\n\n\[ P\left\{ {\mathop{\sup }\limits_{{\left| t\right| < {T}_{N}}}\left| {X\left( {t, N}\right) }\right| \geq v{\Gamma }_{oN}^{1/2}}\right\} \leq C\left( {1 + {T}_{N}{\Gamma }_{1N}/{\Gamma }_{oN}}\right) \exp \left( {-\frac{{v}^{2}}{2 + \varepsilon }}\right) \]\n\n(3.2.11)\n\nfor any \( v > 0 \), w... | Yes |
Theorem 3.2.1 (Csörg \( \delta \), Lin 1990a) Let \( \left\{ {T}_{N}\right\} \) and \( \left\{ {h}_{N}\right\} \) be sequences of positive numbers. Suppose that \( \left\{ {T}_{N}\right\} \) is non-decreasing and \( \left\{ {h}_{N}\right\} \) is monotonic. And suppose that Condition (3.2.1) is satisfied and\n\n\[ \math... | Proof of Theorem 3.2.1.\n\nAt first, we prove that for \( 0 < \varepsilon < 1/2 \)\n\n\[ \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathop{\max }\limits_{{1 \leq n \leq N}}\mathop{\sup }\limits_{{\left| t\right| \leq {TN}}}\mathop{\sup }\limits_{{0 \leq s \leq {h}_{N}}}{\alpha }_{N}\left| {X\left( {t + s, n}\right... | Yes |
Theorem 3.3.1 (Csáki et al. 1990) Assume that for some \( \delta > 0 \)\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }{\gamma }_{k}{\left( \log \left( {\lambda }_{k} \vee e\right) \right) }^{1 + \delta }/{\lambda }_{k} < \infty . \]\n\nThen \( X\left( {t, n}\right) \rightarrow X\left( t\right) \) uniformly in \( t \) ... | The conclusion of Theorem 3.3.1 means that for any \( \varepsilon > 0, T > 0 \) and for almost all \( \omega \in \Omega \) there exists an integer \( {n}_{0} = {n}_{0}\left( {\varepsilon, T,\omega }\right) \) such that\n\n\[ \mathop{\sup }\limits_{{\left| t\right| \leq T}}\left| {X\left( {t, n,\omega }\right) - X\left(... | Yes |
Lemma 3.3.1 Assume that Condition (3.3.2) is satisfied and that \( \sigma \left( \cdot \right) \) defined by\n\n\[ \n{\sigma }^{2}\left( s\right) = E{\left( X\left( t + s\right) - X\left( t\right) \right) }^{2}, \n\]\n\n(3.3.8)\n\n\nis a regular varying function at zero with a positive exponent, namely\n\n\[ \n\sigma \... | Proof For any positive real numbers \( t \) and \( r \) we let \( {t}_{r} = \left\lbrack {{2}^{r}t}\right\rbrack /{2}^{r} \) and write also \( R = {2}^{r} \) . Clearly, using the continuity of \( X\left( \cdot \right) \), we have\n\n\[ \n\left| {X\left( {t + s}\right) - X\left( t\right) }\right| \leq \left| {X\left( {\... | Yes |
Theorem 3.4.1 (Csörgő and Lin 1990b) Assume \( {\Gamma }_{0} < \infty \) and \( {\Gamma }_{2} < \infty \) . Then for any \( {T}_{h} \uparrow \infty \) continuously as \( h \rightarrow 0 \) , \[ \mathop{\lim }\limits_{{h \downarrow 0}}\mathop{\sup }\limits_{{\left| t\right| \leq {T}_{h}}}\mathop{\sup }\limits_{{0 \leq s... | Proof For given \( 0 < \varepsilon < 1 \), let \( {h}_{n} \) be such that \[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{\left( {h}_{n - 1}/{T}_{{h}_{n}}\right) }^{\varepsilon /2} < \infty \] and as \( n \rightarrow \infty \) , \[ \left( {{h}_{n - 1}/{T}_{{h}_{n}}}\right) /\left( {{h}_{n}/{T}_{{h}_{n + 1}}}\right) \right... | Yes |
Lemma 3.4.2 Assume \( {\Gamma }_{0} < \infty \) . Then for any \( \varepsilon > 0 \) we have\n\n\[ P\left\{ {{\chi }^{2}\left( t\right) \geq {2mv}}\right\} \geq \frac{\varepsilon }{6}{v}^{1/2}\exp \left( {-\frac{v}{1 - \varepsilon }}\right) \]\n\nfor any \( t \geq 0 \) and \( v > 0 \) . | Proof (3.4.22) is an immediate consequence of Theorem 1 of Iscoe and McDonald (1989). We only prove (3.4.21).\n\nLet \( {m}_{n} = \mathop{\max }\limits_{{1 \leq j \leq n}}{\gamma }_{j}/{\lambda }_{j},{\tau }_{n} = \mathop{\sum }\limits_{{j = 1}}^{n}{\gamma }_{j}/{\lambda }_{j} \) . At first, we show\n\n\[ P\left\{ {\ma... | Yes |
If \( \Gamma \left( {t, v, x, y}\right) = {I}_{( - \infty, t\rbrack \times \lbrack 0,\underline{v}\rbrack }\left( {x, y}\right) , - \infty < t < \infty \) , \( 0 \leq v < \infty \), then | \[ X\left( {t, v}\right) = W\left( {t, v}\right) , \] \[ {H}_{1}^{2}\left( {t, s, v}\right) = {sv},0 \leq s < \infty , \] \[ {H}_{2}^{2}\left( {t, s, v, u}\right) = {su},0 \leq s, u < \infty . \] | No |
Example 2 If \( \Gamma \left( {t, v, x, y}\right) = {I}_{\left\lbrack {0, t}\right\rbrack \times \left\lbrack {0, v}\right\rbrack }\left( {x, y}\right) - t{I}_{\left\lbrack {0,1}\right\rbrack \times \left\lbrack {0, v}\right\rbrack }\left( {x, y}\right) \) , \( 0 \leq t \leq 1,0 \leq v < \infty \), then | \[ X\left( {t, v}\right) = W\left( {t, v}\right) - {tW}\left( {l, v}\right) \] is a Kiefer process (cf. Section 1.15 in Csörgo-Revesz 1981), | No |
Lemma 3.5.3 Let \( A \subset {R}^{ + },{s}_{0},{u}_{0} > 0 \) . Suppose that\n\n\[ \n{EX}\left( {R\left( {t, s,{v}^{\prime }, b - {v}^{\prime }}\right) }\right) X\left( {R\left( {t, s, v, b - v}\right) }\right) \n\]\n\n(3.5.22)\n\n\[ \n\geq E{\left( X\left( R\left( t, s, v, b - v\right) \right) \right) }^{2} \n\]\n\nfo... | Proof Define\n\n\[ \n{t}_{k + j} = \left( {\left\lbrack {t{2}^{{2}^{k + j}}/{s}_{0}}\right\rbrack + 1}\right) {s}_{0}/{2}^{{2}^{k + j}},{v}_{k + j}^{\prime } = \left( {\left\lbrack {v{2}^{{2}^{k + j}}/{u}_{0}}\right\rbrack + 1}\right) {u}_{0}/{2}^{{2}^{k + j}}. \n\]\n\nNoting that \( X\left( {R\left( {t, s, v, u}\right... | Yes |
Theorem 3. 5. 2 (Csörgő, Lin, Shao 1991) Suppose that ( 3. 5. 22), (3.5.23) and (3.5.24) are satisfied with \( {s}_{0} = {a}_{T},{u}_{0} = {b}_{T} \) and that \[ {EX}\left( {R\left( {{js}, s,{ku}, u}\right) }\right) X\left( {R\left( {{ms}, s,{lu}, u}\right) }\right) \leq 0 \] (3.5.55) for any \( s > 0, u > 0, j \neq k ... | Proof At first, we prove \( \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{\left| t\right| \leq 1}}\mathop{\sup }\limits_{{0 \leq v \leq 1}}\mathop{\sup }\limits_{{0 \leq s \leq {a}_{T}}}\mathop{\sup }\limits_{{0 \leq u \leq {b}_{T}}}\frac{\left| X\left( R\left( t, s, v, u\right) \right) \right|... | Yes |
Let \( \{ W\left( {x, y}\right) ; - \infty < x < \infty ,0 \leq y < \infty \} \) be a standard two-parameter Wiener process. Then \[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{\left| t\right| \leq 1}}\mathop{\sup }\limits_{{0 \leq t \leq 1}}\frac{\left| W\left( R\left( t,{a}_{T}, v,{b}_{T}\r... | \[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{\left| t\right| \leq 1}}\mathop{\sup }\limits_{{0 \leq v \leq 1}}\mathop{\sup }\limits_{{0 \leq s \leq {a}_{T}}}\mathop{\sup }\limits_{{0 \leq u \leq {b}_{T}}}\frac{\left| W\left( R\left( t, s, v, u\right) \right) \right| }{{\left( 2{a}_{T}{b}_{T... | Yes |
Corollary 3.5.2 Let \( \{ K\left( {x, y}\right) ;0 \leq x \leq 1,0 \leq y < \infty \} \) be a Kiefer process. Then | \[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathop{\sup }\limits_{{0 \leq t \leq 1 - {a}_{T}}}\mathop{\sup }\limits_{{0 \leq v \leq 1}}\frac{\left| K\left( R\left( t,{a}_{T}, v,{b}_{T}\right) \right) \right| }{{\left( 2{a}_{T}\left( 1 - {a}_{T}\right) {b}_{T}\log \left( 1/{a}_{T}{b}_{T}\right) \right) }^{1/2}} =... | Yes |
Theorem 1.1.1. For a Gaussian sequence \( \left\{ {{X}_{n}, n \geq 1}\right\} \), we have\n\n\[ \alpha \left( {{\mathcal{F}}_{1}^{k},{\mathcal{F}}_{k + n}^{\infty }}\right) \leq \rho \left( {{\mathcal{F}}_{1}^{k},{\mathcal{F}}_{k + n}^{\infty }}\right) \leq {2\pi \alpha }\left( {{\mathcal{F}}_{1}^{k},{\mathcal{F}}_{k +... | Proof. The former inequality is obvious.\n\nFor any \( \varepsilon > 0 \), there exist two normal random variables \( X \in {L}_{2}\left( {\mathcal{F}}_{1}^{k}\right), Y \) \( \in {L}_{2}\left( {\mathcal{F}}_{k + n}^{\infty }\right) \) such that \( {EX} = {EY} = 0,\operatorname{Var}X = \operatorname{Var}Y = 1 \) and\n\... | Yes |
Theorem 1.1.2. If the spectral function of a stationary sequence is not absolutely continuous, then \( \rho \left( n\right) \equiv 1 \), i.e. the sequence is not \( \rho \) -mixing. Conversely, if the spectral function is absolutely continuous, then\n\n\[ \rho \left( n\right) = \mathop{\inf }\limits_{h}\underset{\lambd... | The Proof of Theorem 1.1.2 is omitted(Kolmogorov, Rozanov 1960). | No |
Lemma 1.2.1. Let \( \\left\\{ {{X}_{n}, n \\in \\mathbb{Z}}\\right\\} \) be an \( \\alpha \) -mixing sequence, \( X \\in {\\mathcal{F}}_{-\\infty }^{k} \) and \( Y \\in {\\mathcal{F}}_{k + n}^{\\infty } \) with \( \\left| X\\right| \\leq {C}_{1} \) and \( \\left| Y\\right| \\leq {C}_{2} \). Then\n\n\[ \n\\left| {{EXY} ... | Proof. By the property of conditional expectation, we have\n\n\[ \n\\left| {{EXY} - {EXEY}}\\right| = \\left| {E\\left\\{ {X\\left( {E\\left( {Y \\mid {\\mathcal{F}}_{-\\infty }^{k}}\\right) - {EY}}\\right) }\\right\\} }\\right|\n\]\n\n\[ \n\\leq {C}_{1}E\\left| {E\\left( {Y \\mid {\\mathcal{F}}_{-\\infty }^{k}}\\right... | Yes |
Lemma 1.2.2. Let \( \\left\\{ {{X}_{n}, n \\in \\mathbb{Z}}\\right\\} \) be an \( \\alpha \) -mixing sequence, \( X \\in {\\mathcal{F}}_{-\\infty }^{k} \) and \( Y \\in {\\mathcal{F}}_{k + n}^{\\infty } \) with \( E{\\left| X\\right| }^{p} < \\infty \) for some \( p > 1 \) and \( \\left| Y\\right| \\leq C \) . Then\n\n... | Proof. Let \( {X}_{N} = {XI}\\left( {\\left| X\\right| \\leq N}\\right) ,{X}_{N}^{\\prime } = X - {X}_{N} \) . Write\n\n\[ \n\\left| {{EXY} - {EXEY}}\\right| \\leq \\left| {E{X}_{N}Y - E{X}_{N}{EY}}\\right| + \\left| {E{X}_{N}^{\\prime }Y - E{X}_{N}^{\\prime }{EY}}\\right| .\n\]\n\nBy Lemma 1.2.1, \( \\left| {E{X}_{N}Y... | Yes |
Lemma 1.2.4. Let \( \\left\\{ {{X}_{n}, n \\in \\mathbb{Z}}\\right\\} \) be an \( \\alpha \) -mixing sequence, \( X \\in {\\mathcal{F}}_{-\\infty }^{k} \) and \( Y \\in {\\mathcal{F}}_{k + n}^{\\infty } \) with \( E{\\left| X\\right| }^{p} < \\infty \) and \( E{\\left| Y\\right| }^{q} < \\infty ,\\frac{1}{p} + \\frac{1... | \[ \\left| {{EXY} - {EXEY}}\\right| \\leq {10}\\parallel X{\\parallel }_{p}\\parallel Y{\\parallel }_{q}{\\left( \\alpha \\left( n\\right) \\right) }^{1 - \\frac{1}{p} - \\frac{1}{q}}. \] | Yes |
Lemma 1.2.5. Let \( \\left\\{ {{X}_{n}, n \\in \\mathbb{Z}}\\right\\} \) be an \( \\alpha \) -mixing sequence, \( X \\in {\\mathcal{F}}_{-\\infty }^{k} \) and \( Y \\in {\\mathcal{F}}_{k + n}^{\\infty } \) with \( E{\\left| X\\right| }^{2 + \\delta } \\leq {C}_{1}, E{\\left| Y\\right| }^{2 + \\delta } \\leq {C}_{2} \).... | \[ \\left| {{EXY} - {EXEY}}\\right| \\leq {10}{\\left( {C}_{1}{C}_{2}\\right) }^{\\frac{1}{2 + \\delta }}{\\left( \\alpha \\left( n\\right) \\right) }^{\\frac{\\delta }{2 + \\delta }}. \] | Yes |
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