91Ó°ÊÓ

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

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.