transformations of random variables and random vectors

... ∈ (P5+)11 and we have a contradiction So [q] 6= 0 and γ1 = 0 The 4 Proof of Theorems 1.2 and 1.3 Proof of Theorem 1.2 Since f Sq0∗ = (fSq0∗)5 15 : (QP5)15 → (QP5)5 is a homomor-phism of GL5-modules, ... 74 x7x2x2x4x4 75 x7x2x2x4x5. Proof Let x be an admissible monomial of degree 15 in P5 and ω(x) = (3, 2, 2) From the proof of Lemma 3.5, x is a permutation of one of the monomials: x1x2x2x4x7, x1x2x3x6x6, ... x3x2x2 40 x7x3x x2x2. Trang 9Proof Let x be an admissible monomial of degree 15 in P5 and ω(x) = (3, 4, 1) From the proof of Lemma 3.5, x is a permutation of one of the monomials x1x2x2x3x7, x1x2x3x3x6,...

Ngày tải lên: 16/10/2015, 09:25

... Usman, 2019) Oil Price has average of US$ 66 and standard of deviation of 23.69 for the period of 2013 – 2018 Inflation has average of 5.05% and Standard of deviation of 1.39% These figures also explain ... statistics of this research variable that is shown by Table below The Effects of Macroeconomic Variables and Company’s Financial Ratios… 173 Stock price has an average return of 9,45% with standard of ... Member of Research Cluster of Financial Institution at FIA UI Professor of Administrative Science, Head of Research Cluster of Financial Institution at FIA University Indonesia Faculty of Administration...

Ngày tải lên: 01/02/2020, 22:42

... array of rowwise ρ-mixing random variables satisfying 2/q supn ∞1 ρn 2i < ∞ for some q ≥ and EXni for all n ≥ 1, i ≥ Let the random variables in i each row be stochastically dominated by a random ... dependent random variables,” Theory of Probability and Its Applications, vol 20, pp 134–139, 1975 M Peligrad, “On the central limit theorem for ρ-mixing sequences of random variables,” The Annals of ... sequences of random variables,” The Annals of Probability, vol 10, no 4, pp 968–981, 1982 Q M Shao, “A remark on the invariance principle for ρ-mixing sequences of random variables,” Chinese Annals of...

Ngày tải lên: 21/06/2014, 07:20

... function F(x) of the Xj should belong to the domain of attraction of the stable law, and (2) there exists N with sup p N (x) < oo Proof. Condition (2) is clearly necessary, and implies that ... ap (x) Now let X1, X2 , be a sequence of independent random variables with distribution F, and denote as before by Fn(x) the distribution function of the normalised sum Z,, = (X1 + X2 + ... of some stable distribution G with exponent a (0<a<,2), it is necessary and sufficient that (1) the common distribution function F of the X X should belong to the domain of attraction of...

Ngày tải lên: 02/07/2014, 20:20

... -1 ) (5 3.17) The proof of 82 = o(n -1 ) is similar,... 27r/h and define S 1 (x) and d,,(t) as in § 3.3 Theorem 5. 3 2 If X1 , X2 , are independent random variables with the same ... belonging to L h and having finite third moment, then, for 1, 1 then, as n-+ oo, IIFn - GnIIp = o(n - gyp) (5. 3 12) We are now in a position to complete the proof of the theorem If ... f T eitX d { (27rn) Z dt=2(8 1 t +82) , and eit"dHH (x)= hn (t) = co 00 e`t" dGn (x) cc + J S 1 (x) e ZXz} = gn W+ do (t) Exactly as in the proof of Theorem 3 3.2, we have S = o(n -1 )...

Ngày tải lên: 02/07/2014, 20:20

... such variables those of class (C), and distinguish the subclass (C, d) of variables with a bounded continuous probability density g (x), and the subclass (C, e) of lattice variables, i e those ... THEOREMS AND BERNSTEIN'S INEQUALITY „ 1 Statement of the theorems The theorems of this chapter do not have a collective character, and are related to Theorem 6 1 1 We shall consider a sequence of ... for lattice variables We now proceed to the proof of Theorem 7 1 2 We introduce 00 M (z) = E(ez Xi) = Pkexp[(kh + b) z] , (7.4 1) k=-co defined in IRe zi < a because of (7 1 2), and periodic...

Ngày tải lên: 02/07/2014, 20:20

... 1 and (9 2 5) contradicts (9 2 4) The case of (9 2 3)istreated similarly Theorem 9 2 2 For random variables of class (d) the condition (9 2 1) is necessary in order that [0, n" p (n)] and ... 9MONOMIAL ZONES OF LOCAL NORMAL A 11 RACTION „ 1 Zones of normal attraction In this chapter it will be assumed that the independent random variables X; satisfy E(X;)=0, V(XX)=62>0, and that Sn ... ] and [ - n" p (n), 0] to be zones of local normal attraction, is also sufficient for [0, n"/p (n)] and [ - n"/p (n), 0] to be zones of local normal attraction If on the other hand...

Ngày tải lên: 02/07/2014, 20:20

... exp(2XP(n))} (11 9 10) Trang 1411 10 COMPLETION OF THE PROOF OF THEOREM 11 2 1 A similar analysis can be made of „ 10 Completion of the proof of Theorem 11 2.1 We now investigate, for 3 < ... Trang 2„ 2 Statement of the theoremsWe shall investigate the narrow zones of local and integral normal vergence for variables of the class (d)defined in Chapter 9 In terms of thefunction h (x) ... and [- nZ- R/p 1 (n) ] are zones of localnormal attraction, and completes the proof of Theorem 11 2 1 „ 11 The corresponding integral theorem Consider first the monomial zones [0, n"] and...

Ngày tải lên: 02/07/2014, 20:20

... the methods of Chapter 9 „ 2 An upper bound for the probability of a large deviation We now proceed to the proof of the sufficiency part of Theorem 12 1 2, assuming (12 1 3) and 227 and an analogous ... terms of its predecessors In this chapter we prove the following theorems Theorem 12 1 1 If [ - n", 0] and [0, n"] are zones of normal attraction for all a < 2, then the variables ... PROBABILITY OF A LARGE DEVIATION I n order that [0, no'p (n)] and [ - no 'p (n), 0] be zones of normal attraction, it is necessary that E(exp A IXil4a/(2a+ 1)) < oc (0 < A < 1) , (12 1 3) and...

Ngày tải lên: 02/07/2014, 20:20

... shows that the members of Ko all have size > m Thus the number of members of Ko cannot exceed the number of subsets of 2C of size m, which is (n,) Lemma 15 2 2 If the random variables Xl in Theorem ... (15 2 1) The proof of this theorem requires a number of auxiliary results Lemma 15.2 1 Let 21 be a set of n elements, and K a class of subsets of 21 such that no member of K is contained ... function of a random variable X is the function Q X(l) = Q(l) =sup P(x<X <x+l) X As a function of 1>0, this is non-decreasing and right-continuous If X, Y are independent random variables,...

Ngày tải lên: 02/07/2014, 20:20

... probability 1, of the random variables ~n j x C there exists one and only one transformation Ti (up to events of probabili-ty 0), satisfying the conditions (1 1 )-{4 1 ) If now c is any random variable ... subset of the real line) The random process is called stationary (in the wide sense) ifE (Xr) < 00 for all t, and ifE(XX) and E(XSXX+t) do not depend on s Without loss of generality we can (and ... values of t are considered, a semigroup of isometric operators) In the discrete time case, this is the cyclic group of powers of the restriction U of Ti In fact, because of (16 2 3) and (16...

Ngày tải lên: 02/07/2014, 20:20

... distributed random variables is regular In the Hilbert space terminology of the last chapter, regularity simply means that the subspace H_~=H_~=n H`_ 00 t (which consists of the random variables ... ; )of Yis the sum of the spectral functionFy( N ) ( ?) ofY(N) and Fz( N) (A)of Z(N) The process Y1 N 'is linearly regular and Fy(N) (A) is absolutely continuous Thus the total variation of ... magnitude of the left-hand side measures the de-pendence between past and future, and it may be useful to assume this to be small, in some sense In this chapter we examine some of the possible ways of...

Ngày tải lên: 02/07/2014, 20:20

... (t)dt , (18 3.7) 0 and since R (t) is continuous, t -1 ~ limEli X(t)dt-X(0) _ -r-0 0 t t... distribution of Z n is the same as that of Z;,, to the investigation of which we now turn ... function of an 1 ~ o by /n (t), and prove that IE(e On -+ 0 (18. 4.12) "z") (t)k I as n > oo The variable exp it (~o+ +~k-2) n is measurable with respect to 9(k -1 )p+(k-2)qI and it ... (1+E3)s < c4(1+E3) 1-1 If E 3 and e are chosen so small that (1+E3)(2+E) 2-1 -+ b < p < 1 , we obtain Yr < c5/(1 - P) = c3'' Thus, for this choice of e, 18 5 SUFFICIENT CONDITIONS...

Ngày tải lên: 02/07/2014, 20:20

... number of states(labelled 1, 2, , k) and transition matrix P = (p i ;) (see, for instance,Chapter III of [47] ) If Xn is the state of the system at time n, we have thesequence of random variables ... distribution of values of sums of the form Ef (2kx) Letf (t) be a periodic function of the real argument t, with period 1, andconsider the distribution of the values of the sum Such sums are of considerable ... metric theory of continued fractions Each real number t in the interval (0, 1) has a unique continued fractionexpansion of the form of computing the measures of sets of values of t defined by...

Ngày tải lên: 02/07/2014, 20:20

... sums of random variables defined on a finite homo-geneous Markov chain © 2 : m-dependent random variables were first studied by Hoeffding andRobbins [51] Theorem 19 2 1 is due to Diananda [25], ... (7) The discovery of wider classes of random variables, for whichintegral theorems are valid on the whole line (8) As a particular case of (7), study the random variables with uous probability ... case p=2, the point of view of © 16 3 gives (A3 1) a simple geometricmeaning IfHn is the subspace of L 2 (Q)consisting of the random variablesmeasurable with respect to Di n , and Pn is the projection...

Ngày tải lên: 02/07/2014, 20:20

... particular class of random variables, called circular complex random variables Circularity is a type of symmetry in the distributions of the real and imaginary parts of complex random variables and stochastic ... “Complex Random Variables and Stochastic Processes”Digital Signal Processing Handbook Ed Vijay K Madisetti and Douglas B Williams Boca Raton: CRC Press LLC, 1999 Trang 260 Complex Random Variables and ... center frequency of the bandB, and “LPF” represents an ideal lowpass filter whose bandwidth is greater than half the bandwidth ofs(t), but much less than 2ω c The Fourier transform ofg(t) is given...

Ngày tải lên: 25/10/2013, 02:15

... (Electronic) Random Vectors and Independence In this chapter, we review central concepts of probability theory,statistics, and random processes The emphasis is on multivariate statistics and random vectors ... probability, random variables, and stochastic processes is [353] 2.1 PROBABILITY DISTRIBUTIONS AND DENSITIES 2.1.1 Distribution of a random variable In this book, we assume that random variables ... gaussian random vector that has the mean vector x = ( 3)T and covariance matrix m Cx = 2 2.15 Assume that random variables x and y are linear combinations of two uncorrelated gaussian random variables...

Ngày tải lên: 13/12/2013, 14:15

... number of cations and anions + where v c and w h ile p and pa are the chemical potential of the cation respectively T h u s t h e condition of equilibrium for a strong dissolved in water and in ... Another colleague of Or Zemaitis, Dr Marshall Rafal, owner of OLI Systems, Inc., assumed the contractor’s responsibility for execution of the handbook and contributed some of the writing Dr Scrivner ... equation ( ) sum of cation and anion stoichiometric numbers harmonic mean of v + and v-, equation ( ) vi - stoichiometric number of ion i n P - T - NWTL binary interaction parameter of Chen's activity...

Ngày tải lên: 01/04/2014, 11:38