Question

Prove that the sequence of random variables Zn in Exercise 22 converges in quadratic mean (definition in Exercise 10) to 0.

Short answer

It is proved that the sequence of random variables converges in quadratic mean to 0.

Step-by-step solution

Given information

Random variables have a uniform distribution on the interval\(\left[ {0,1} \right]\).

Explanation

Let\({Z_n} = {h_n}\left( X \right)\).

Each function\({h_n}\)will take only two values, 0 and 1.

The set x where\({h_n}\left( x \right) = 1\)is determined by dividing the interval\(\left[ {0,1} \right]\)into k nonoverlapping subintervals of length\(\frac{1}{k}\)for\(k = 1,2,...\)

Assume that the sequence of random variables converges in quadratic mean to 0.

It said that a sequence of random variables\({Z_1},{Z_2},...\)converges to a constant b in quadratic mean if

\(\mathop {\lim }\limits_{n \to \infty } E\left[ {{{\left( {{Z_n} - b} \right)}^2}} \right] = 0\)………………… (1)

Each\({Z_n}\)has the Bernoulli distribution with parameters\(\frac{1}{{{K_n}}}\). Hence from equation (1)

\(E\left[ {{{\left( {{Z_n} - 0} \right)}^2}} \right] = \frac{1}{{{k_n}}}\)which goes to 0.

Hence, the sequence of random variables converges in quadratic mean to 0.