Assume that V is the set of all complex sequences, (xn), that satisfy the relation Xn+nXn+1...
5. Let V = Mn,n(C) (the vector space of nxn complex matrices). Let Sy be the set of all Hermitian matrices in V, and let Sy be the set of all unitary matrices in V. Are SH and/or Su subspaces of V?
Let V be the vector space of all sequences over R. Given (a1, a2, T,U V V by ) e V, define : ) ...) = (0, a1, 0, a2, 0, a3, . . . ) Тај, а2, аз, ад, 0, аз, (a1, a3, a5,.) and U(a1, a2, a3, a4, (a) Find N(T) and N(U) (b) Explain why T is onto, but not 1-1 (c) Explain why U is 1-1, but not onto.
Let Z1, Z2, . . . be a sequence of independent standard normal random variables. Define X0 = 0 and Xn+1 = (nXn + (Zn+1))/ (n + 1) , n = 0, 1, 2, . . . . The stochastic process {Xn, n = 0, 1, 2, } is a Markov chain, but with a continuous state space. (a) Find E(Xn) and Var(Xn). (b) Give probability distribution of Xn. (c) Find limn→∞ P(Xn > epsilon) for any epsilon > 0.
the set of compactly supported sequences is defined by c00 = {{xn} : there exists some N ≥ 0 so that xn = 0 for all n ≥ N } Prove that for 1 ≤ p ≤ ∞ the metric space (c00, dp) is not complete.
1. [4-+6+6-16 points Let /°0 denote the vector space of bounded sequences of real numbers, with addition and scalar multiplication defined componentwise. Define a norm Il on by Il xl = suplx! < oo where x = (x1,x2, 23, . .. ) iEN (a) Prove that is complete with respect to the norm | . (b) Consider the following subspaces of 1o i) c-the space of convergent sequences; (i) co-the space of sequences converging to 0; (iii) coo- the space...
real analysis
hint
9 Let co , a, and 〈æ be the Banach spaces consisting of all complex sequences x={ i-1, 2, 3,..., defined as follows: X E if and only if II x11 if and only if lxsup lloo. for which ξί (a) If y = {nJ E 11 and Ax = Σ ζίηǐ for every x ε co, then Λ is a bounded linear functional on (More precisely, these two spaces are not equal; the preceding statement exhibits...
2.13.4 2.13.5 Show that lim supno (-X) = -(liminf ,-Xn). If two sequences {an) and {bn} satisfy the inequality an <b, for all sufficiently large n, show that limsupan Slim sup bn and liminfa, <liminf bn. 100 2.13.6 Show that lim, 100 Xn = o if and only if lim sup.Xn = liminf xn = c. n-00 2.13.7 Show that if lim sup a n = L for a finite real number L and € > 0, then an >...
Theorem 7.3.5 Let P be a partition of a nonempty set X. Define a relation~on X for all a, b X by defining: Then is an equivalence relation on X. Furthermore, the equivalence classes ofare exactly the elements of the partition P: that is, X/ ~= P. Proof: See page 164 in your textbook. a,b,c,d,e,f partition P = {{a, c, e), {b, f}, {d)) 5 Let A = Give a complete listing of the ordered pairs in the equivalence relation...
Please answer all the parts neatly with all details.
3. Assume X1, X2,... are a sequence of i.i.d. random variables having finite first moment, that is, v = E(Xi oo. Let Yn = (|X1| .+ |Xn|)/n. (a) Show that Yn ->v in probability. (b) Show that E(Y,) -- v. (c) Show that E(|X, - /u|) -0 where u = E(X)
3. Assume X1, X2,... are a sequence of i.i.d. random variables having finite first moment, that is, v = E(Xi...
ANSWER 1 & 2 please. Show work for my understanding and
upvote. THANK YOU!!
Problem 1. Let {x,n} and {yn} be two sequences of real numbers such that xn < Yn for all n E N are both convergent, then lim,,-t00 Xn < lim2+0 Yn (a) (2 pts) Prove that if {xn} and {yn} Hint: Apply the conclusion of Prob 3 (a) from HW3 on the sequence {yn - X'n}. are not necessarily convergent we still have: n+0 Yn and...