Let xr be a sequence of random(n x 1) vector such that for some b>0. If g(x)is a real-valued function with gradient g' (a)(ag then T(g(xr)-g(a))-g(a)x Example Let Y, Y2, ,Yr be an ii d sample of size T deawn from a distribution with mean u+0 and variance o. Consider the distribution of the reciprocal of the sample mean, ST=1/ YT, where Yr=(1/T)2Y. We know from the CLT that VT(Yr -u)Y, where Y NN(0, o2). Also, g(y)=1/y is continous at y=u Let g(u(ag/ayly=u)=(1/u2). Then VTIST-(1/)1-g(u)r in other word, VTIST-(1/p)I-N(0,02/u)Let {xT } be a sequence of random (n × 1) vector such that T b (xT − a) L−→ x for some b > 0. If g(x) is a real-valued function with gradient g 0 (a)(= ∂g ∂x0   x=c ), then T b (g(xT) − g(a)) L−→ g 0 (a)x. Example: Let {Y1, Y2, ..., YT } be an i.i.d. sample of size T deawn from a distribution with mean µ 6= 0 and variance σ 2 . Consider the distribution of the reciprocal of the sample mean, ST = 1/Y¯ T , where Y¯ T = (1/T) PT t=1 Yt . We know from the CLT that √ T(Y¯ T − µ) L−→ Y , where Y ∼ N(0, σ 2 ). Also, g(y) = 1/y is continous at y = µ. Let g 0 (u)(= ∂g/∂y|y = µ) = (−1/µ2 ). Then √ T[ST − (1/µ)] L−→ g 0 (µ)Y ; in other word, √ T[ST − (1/µ)] L−→ N(0, σ 2/µ4 ). 9