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If (xn)– is a convergent sequence with limn700 Xn = 0 prove that x1 + x2+...+xn...
Let X1, X2,...be a sequence of random variables. Suppose that Xn?a in probability for some a...
Let X1, X2,...be a
sequence of random variables. Suppose that Xn?a in probability for
some a ? R. Show that (Xn) is Cauchy convergent in probability,
that is, show that for all
> 0 we have P(|Xn?Xm|> )?0 as n,m??.Is the converse true?
(Prove if “yes”, find a counterexample if “no”)
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Form...
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
Problem 10. We define a sequence {xn} in R by x1 = 1, Xn+1 = 1...
Problem 10. We define a sequence {xn} in R by x1 = 1, Xn+1 = 1 1+ Xn Prove that {xn} is convergent and find the limit.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence...
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
Let X = x1, x2, . . . , xn be a sequence of n integers....
Let X = x1, x2, . . . , xn be a sequence of n integers. A sub-sequence of X is a sequence obtained from X by deleting some elements. Give an O(n2) algorithm to find the longest monotonically increasing sub-sequences of X.
13 14 Exercise 13: Let (xn) be a bounded sequence a S be the set of...
13 14
Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
Find the extreme values of f f(x1,x2,...,xn) = x1x2...xn(1 − x1 − 2x2 − ··· − nxn), if x1 > 0,...
find the extreme values of f f(x1,x2,...,xn) = x1x2...xn(1 − x1 − 2x2 − ··· − nxn), if x1 > 0, x2 > 0, ...,xn > 0.
Prove that a sequence of random variables X1, X2, ... converges in probability to a constant...
Prove that a sequence of random variables X1, X2, ... converges in
probability to a constant μ if and only if it also converges in
distribution to μ.
5. Prove that a sequence of random variables X1, X2,... converges in probability to a constant p if and only if it also converges in distribution to u.
ANSWER 1 & 2 please. Show work for my understanding and upvote. THANK YOU!! Problem 1....
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...
Let X1, X2,...be a
sequence of random variables. Suppose that Xn?a in probability for
some a ? R. Show that (Xn) is Cauchy convergent in probability,
that is, show that for all
> 0 we have P(|Xn?Xm|> )?0 as n,m??.Is the converse true?
(Prove if “yes”, find a counterexample if “no”)
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
Problem 10. We define a sequence {xn} in R by x1 = 1, Xn+1 = 1 1+ Xn Prove that {xn} is convergent and find the limit.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0.
Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
13 14
Exercise 13: Let (xn) be a bounded sequence a S be the set of limit points of (n), i.e. S:{xER there exists a subsequence () s.t. lim } ko0 Show lim inf inf S n-o0 Hint: See lecture for proof lim sup Exercise 14: (Caesaro revisited) Let (x) be a convergent sequence. Let (yn) be the sequence given by Yn= n for all n E N. Show that lim sup y lim sup n n-+00 n o0
Prove that a sequence of random variables X1, X2, ... converges in
probability to a constant μ if and only if it also converges in
distribution to μ.
5. Prove that a sequence of random variables X1, X2,... converges in probability to a constant p if and only if it also converges in distribution to u.
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...
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