whereas a sequence i lg-o is said to be absolute-summable if pil< It is important to note that absolute summability implies square-summability, but the converse does not hold Proposition If the coefficients of the M A(oo)is square-summable, then iso pjiEt-iconverges in mean square to some random variable Yt as T-o0 Proof The Cauchy criterion states that 2io; converges in mean square to some random variable Yt as T-o if and only if, for any s>0, there exists a suitably arge N such that for any integer M>N jeT In words, once N terms have been summed, the difference between that sum and the one obtained from summing to m is a random variable whose mean and variance are both arbitrarily close to zero Now the left hand side of (1)is simply EIPMEt-M+PM-lEt-M+1+.+PN+1Et-N-1 But if 2i=oP; oo, then by the Cauchy criterion the right side of (2)may be made as small as desired by a suitable large N. Thus the M A(oo)is well defined sequence since the infinity series 2i=0; converges in mean squares 1.3.2 Check Stationarity Assume the MA(oo) process to be with absolutely summable coefficientswhereas a sequence {ϕj} ∞ j=0 is said to be absolute-summable if X∞ j=0 |ϕj | < ∞. It is important to note that absolute summability implies square-summability, but the converse does not hold. Proposition: If the coefficients of the MA(∞) is square-summable, then P∞ j=0 ϕjεt−j converges in mean square to some random variable Yt as T → ∞. Proof: The Cauchy criterion states that P∞ j=0 ϕjεt−j converges in mean square to some random variable Yt as T → ∞ if and only if, for any ς > 0, there exists a suitably large N such that for any integer M > N E "X M j=0 ϕjεt−j − X N j=0 ϕjεt−j #2 < ς. (1) In words, once N terms have been summed, the difference between that sum and the one obtained from summing to M is a random variable whose mean and variance are both arbitrarily close to zero. Now the left hand side of (1) is simply E [ϕMεt−M + ϕM−1εt−M+1 + .... + ϕN+1εt−N−1] 2 = (ϕ 2 M + ϕ 2 M−1 + ... + ϕ 2 N+1)σ 2 = "X M j=0 ϕ 2 j − X N j=0 ϕ 2 j #2 σ 2 . (2) But if P∞ j=0 ϕ 2 j < ∞, then by the Cauchy criterion the right side of (2) may be made as small as desired by a suitable large N. Thus the MA(∞) is well defined sequence since the infinity series P∞ j=0 ϕjεt−j converges in mean squares. 1.3.2 Check Stationarity Assume the MA(∞) process to be with absolutely summable coefficients. 5