(b) EXTeL<∞,i=1,2,…,k; (a)EX|2<∞,i=1,2,…k; (b)M=E(xtxt) is positive definite Then→. R 1. Assumption(2a)is talking about of the mean of this i i.d. sequences(Xtet, i= 1, 2,.,k), see Proposition 3.3 of White, 2001, p. 32) and(2b)is about its first moment exist 2. Assumption(3a) guarantee its(Xti Xti) first moment exist by Cauchy-Schwarz inequality and (3b)is talking about of the mean of this i i d (Xti Xti, i=1, 2, k;j= 1,2,…,k) sequence. An existence of the first moment is what is need for Lln of i.i. d. sequence See p. 15 of Ch 4 It is obvious that from these assumptions we have XtEt E and →E∠t=1Xtx M Therefore 月一B.(b) E|Xtiεt | < ∞, i = 1, 2, ..., k; (3). (a) E|Xti| 2 < ∞, i = 1, 2, ..., k; (b) M ≡ E(xtx 0 t ) is positive definite; Then βˆ a.s −→ β. Remark: 1. Assumption (2a) is talking about of the mean of this i.i.d. sequences (Xtiεt , i = 1, 2, ..., k), see Proposition 3.3 of White, 2001, p.32) and (2b) is about its first moment exist. 2. Assumption (3a) guarantee its (XtiXtj ) first moment exist by Cauchy-Schwarz inequality and (3b) is talking about of the mean of this i.i.d. (XtiXtj , i = 1, 2, .., k; j = 1, 2, ..., k) sequence. An existence of the first moment is what is need for LLN of i.i.d. sequence. See p.15 of Ch.4. Proof: It is obvious that from these assumptions we have  X0ε T  = PT t=1 xtεt T ! a.s −→ E PT t=1 xtεt T ! = 0 and  X0X T  = PT t=1 xtx 0 t T ! a.s −→ E PT t=1 xtx 0 t T ! = M. (2) Therefore βˆ − β a.s −→ M−10 = 0, or βˆ a.s −→ β. 2