Convergencetoa ConstantSeguences and limits.Sequence of constants, indexed by n(n(n+1)/2 + 3n + 5)Ordinary limit:?(n2 + 2n + 1)(The use of the "leading term')Convergence of a random variable. What does itmean for a random variable to converge to a constant?Convergence of the variance to zero. The randomvariable converges to something that is not random
Convergence to a Constant Sequences and limits. Sequence of constants, indexed by n (n(n+1)/2 + 3n + 5) Ordinary limit: - ➔ ? (n2 + 2n + 1) (The use of the “leading term”) Convergence of a random variable. What does it mean for a random variable to converge to a constant? Convergence of the variance to zero. The random variable converges to something that is not random
ConvergenceResultsConvergence of a seguence of random variables to a constant -convergenceinmeansguare:Meanconvergestoaconstant, variance convergesto zero.(Farfrom the mostgeneral, but definitely sufficient for our purposes.)x, =nx, E[x,]=μ→μ, Var[x,]=? / n→0Aconvergencetheoremforsamplemoments.Samplemomentsconvergeinprobabilitytotheirpopulationcounterparts.Generallytheform of TheLawof LargeNumbers.(Manyforms;seeAppendixD inyour text.)Note the great generality of the preceding result. (1/n)Z,g(z)converges to E[g(z.)]
Convergence Results Convergence of a sequence of random variables to a constant - convergence in mean square: Mean converges to a constant, variance converges to zero. (Far from the most general, but definitely sufficient for our purposes.) A convergence theorem for sample moments. Sample moments converge in probability to their population counterparts. Generally the form of The Law of Large Numbers. (Many forms; see Appendix D in your text.) Note the great generality of the preceding result. (1/n)Σig(zi ) converges to E[g(zi )]. 1 2 1 , [ ] , Var[ ]= / 0 n n i i n n n x x E x x n = = → → =
ProbabilityLimitLet be a constant, be any positive value,and n index the sequence.If lim(n -→ 0)Prob[Ix, -0/ > s] = 0 then, plim ×, = 0.X, converges inprobability to 0.In words, the probability that the differencebetween x. and 0 is larger than for any 8goes to zero. x, becomes arbitrarily close to 0.Mean square convergence is sufficient (not necessary)for convergence in probability. (We will not requireother, broader definitions of convergence, such as"almost sure convergence
Probability Limit → − = = n n n Let be a constant, be any positive value, and n index the sequence. If lim(n )Prob[|x | > ] 0 then, plim x . x to . In words, the probability that the difference betwe converges in probability n n en x and is larger than for any goes to zero. x becomes arbitrarily close to . Mean square convergence is sufficient (not necessary) for convergence in probability. (We will not require other, broader definitions of convergence, such as "almost sure convergence
MeanSguare Convergencen = 1000n=100n=10EstimatorFIGURED.1Quadratic ConvergencetoaConstant,e
Mean Square Convergence
ProbabilityLimitsand ExpecationsWhat is the difference betweenE[xn] and plim X,?A notationP>0plim x, = 0Xp
Probability Limits and Expecations What is the difference between E[xn] and plim xn? P n n A notation plim x x = ⎯⎯→