where th is point that lies on the line segment joining thand th since this unique global minimumthere must be small enough neighborhood of thsay thsuch that is strictly (also known as stronglyconvex function on th in other wordsh(this positive definite matrix for all th th and there exists smallest positive eigenvalue such that xh(th) kxk for all in additionsince the maximum eigenvalue of (this continuous function of th th and th is boundedthere must exist constant such that xh(th) kxk for all in other wordsdenoting : (th)we have the boundsl kth thk -( (thr kth thk th th thereforee - rz th nl kth-thk (the dth th - (thg(the - rdth th nl (the kth-th dth an application of theorem yields th (the- (thdth ( - / / andmore importantlyz ln (the- (thdth - rln th - (thp thusthe proof will be complete once we show that th (the dthwith th : this asymptotically negligible compared to th (the- (thdth since th is global minimum that lies outside any neighborhood of ththere must exists constant such that (th)-rc for all th th thereforez - (th-( - rg(the dth (the- (the-( - )( (th)- dth th zth -( - (the- (the-( - ) dth th -( - )( + (the- (thdth ( - ( +cr rp the last expression is of order ( - / / )concluding the proof markov chains definition markov chain markov chain is collection {xt of random variables (or random vectorswhose futures are conditionally independent of their pasts given their present values that isp[xt+ tp[xt+ xt for all markov chain ( in other wordsthe conditional distribution of the future variable xt+ given the entire past { }is the same as the conditional distribution of xt+ given only the present xt property ( is called the markov property markov property