Question: Suppose that X 1 , X 2 , ... is a sequence of random variables, and R and > 0 are constants such that n

Suppose that X1,X2,... is a sequence of random variables, and R and >0 are constants such that n(Xn) converges in distribution to N(0,2).

Let f:RR be a differentiable function such that f is continuous at . Prove that n(f(Xn)f()) converges in distribution to N(0,f()22). (This is known as the 'delta method'.)

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Prove Theorem 3.3 using the theorems of Borel and Cantelli. That is, let $\left\{X_{n}ight\}_{n=1}^{\infty}$ be a sequence of random variables that converges completely to a random variable $X$...

18.2 Suppose that An is a sequence of random variables that converges in probability to 3. Suppose that Bn is a sequence of random variables that converges in distribution to a standard normal. What...

Let Xn be a sequence of random variables that converges in distribution to a random variable X. Let Yn be a sequence of random variables with the property that for any finite number c, Show that for...

Hello! Hope all is well. This is a probability theory class(masters level). I need a correct detailed answer for this question please! 1. Let (Xn) be a sequence of random variables on a probability...

14. Let (Xn : n 1) be a sequence of random variables which converges in mean square. Show that E ???? [Xn Xm]2 0 as m, n . If E(Xn) = and var(Xn) = 2 for all n, show that the correlation between...

Show that a sequence of random variables x1,x2,...xn converges to an indicator probability model, if the distribution function (CDF) of each x is given by: 0 1

Given X 1 , X2 , X3 , X 4 are independent random Variables , each with density function f (a ) = S - ( x - 6 ) if x 7 6 0 else CDF Of f (2) is F (x ) = px Let u = 6- 2 6 du = -1 => da = -du dx = -...

Let 1 and 2 be some unobserved Bernoulli random variables and let x be an observation. Conditional on x = x , the posterior joint PMF of 1 and 2 is given by | ) ( x | ) = { 0 . 2 6 , i f 1 = 0 , 2 =...

PROBABILITY . The bank safe code contains 2 Latin letters and 4 numbers. What is the probability of opening the safe by dialing the code at random if it is known that 1) neither letters nor numbers...

2.9 Suppose that for each 1 j k, n,j , n = 1, 2, . . . , is a sequence of random variables such that n,j P 0 as n. Define n = max1jk |n,j |. (i) Show that if k is fixed, then n P 0 as n. (ii) Give...

Measuring specimen of nylon yarn from two spinning machines. It was found that 8 specimens from the Ist machine have a mean diameter of 9.67 with standard deviation 1.81, while 10 specimens from the...

Figure shows a stationary arrangement of two crayon boxes and three cords. Box A has a mass of 11.0 kg and is on a ramp at angle = 30.0?; box B has a mass of 7.00 kg and hangs on a cord. The cord...

QUESTION 9 Which one of the following matations in the primary struclure would you expect to cause tre most drastic atteration in the finctional pertornance of a protern? Replacement of Val with Ala...

A company is preparing completing their Cash Budget. The following data has been prepared for cash receipts and payments. January February March Cash receipts $1,061,200 $1,182,400 $1,091,700 Cash...