1.4 Convergence in rth mean A stronger condition than convergence in probability is mean square convergence Definition Let (brl be a sequence of real-valued random variables such that for some r>0 r<∞. If there exists a real number b such that e(|br-b)→0asT→∞, then br converge in the rth mean to b, written as br - b The most commonly encountered situation is that of in which r=2. in which case convergence is said to occur in quadratic mean, denoted br =b, or con- vergence in mean square, denoted br - b Proposition(Generalized Chebyshev inequality) Let Z be a random variable such that EZ< oo, r>0. Then for every E>0, Pr(z>e EZr When r= l we have markov's inequality and when r=2 we have the familiar Chebyshev inequality Theorem. If br -b for some r>0 then br -P,b Proof: Since e(b-b)→0asT→∞,E(|br-b)< oo for all T sufficiently large It follows from the Generalized Chebyshev inequality that, for every E>0 Pr(s: or(s)-bl>e)<Elbr-bI- ence br(s)-b<e)≥1-→1asT→∞, since br"b.It follows that Without further conditions, no necessary relationship holds between conver- gence in the rth mean and almost sure convergence1.4 Convergence in rth mean A stronger condition than convecrgence in probability is mean square convergence. Definition: Let {bT } be a sequence of real-valued random variables such that for some r > 0, E|bT | r < ∞. If there exists a real number b such that E(|bT −b| r ) → 0 as T → ∞, then bT converge in the rth mean to b, written as bT r.m. −→ b. The most commonly encountered situation is that of in which r = 2, in which case convergence is said to occur in quadratic mean, denoted bT q.m. −→ b, or con￾vergence in mean square, denoted bT m.s −→ b. Proposition (Generalized Chebyshev inequality): Let Z be a random variable such that E|Z| r < ∞, r > 0. Then for every ε > 0, Pr(|Z| > ε) ≤ E|Z| r ε r . When r = 1 we have Markov’s inequality and when r = 2 we have the familiar Chebyshev inequality. Theorem: If bT r.m. −→ b for some r > 0, then bT p −→ b. Proof: Since E(|bT − b| r ) → 0 as T → ∞, E(|bT − b| r ) < ∞ for all T sufficiently large. It follows from the Generalized Chebyshev inequality that, for every ε > 0, Pr(s : |bT (s) − b| > ε) ≤ E|bT − b| r ε r . Hence Pr(s : |bT (s) − b| < ε) ≥ 1 − E|bT −b| r ε r → 1 as T → ∞, since bT r.m. −→ b. It follows that bT p −→ b. Without further conditions, no necessary relationship holds between conver￾gence in the rth mean and almost sure convergence. 6