Chapter 6 Large Random Samples

6.1 Introduction

It is said that a sequence of random variables \(X_i\) converges to the constant \(b\) in quadratic mean if \[\lim_{n\to\infty}E\left[ (X_n-b)^2 \right] = 0\] Show this equality is satisfied if and only if \[\lim_{n\to\infty} E(X_n) = b \;\;\;\; \textrm{ and } \;\;\;\; \lim_{n\to\infty} Var(X_n)=0\] Hint: \(E[ (Y-c)^2 ] = (\mu -c)^2 + \sigma^2)\) where \(\mu=E(Y)\) and \(\sigma^2 = Var(Y)\).
Prove that if a sequence of random variables \(Z_n\) converges to a constant \(b\) in quadratic mean, then the sequence also converges to \(b\) in probability.