the probability of observing a realization of (Zt) for which convergence to b does not occur is zero. Failure to converge is possible but will almost never happen under this definition Proposition: Given g: RKR(k, I E M and any sequence of random k x 1 vector bT such that br b, where b is k x 1, if g is continous at b, then g(br)gb) This results is one of the most important in this Chapter, because con esults for many of our estimators follows by simply applying this Proposition 1.3 Convergence in Probability a weaker stochastic convergence concept is that of convergence in probability Definition Let br be a sequence of real-valued random variables. If there exists a real num- ber b such that for every6>0, such that Pr(s:|br(s)-b<0)→1,asT→∞ then br converge in probability to b, written as br- b or plim br= b Example Let Zr=T- Eta Zt, where [Zt) is a sequence of random variables such that E(Zt=u, Var(Zt)=02<o for all t and Cou(Zt, Z)=0 fort+T.Then Zr-u by the Chebyshev weak law of large numbers. See the plot of Hamilton 184 When the plim of a sequence of estimator(such as iZrlt_) is equal to the true population parameter(in thius case, p), the estimator is said to be consistent Convergence in probability is also referred as weak consistency, and since this has been the most familiar stochastic convergence concept in econometrics the word"weak"is often simply droppedthe probability of observing a realization of {Zt} for which convergence to b does not occur is zero. Failure to converge is possible but will almost never happen under this definition. Proposition: Given g : Rk → Rl (k, l ∈ N ) and any sequence of random k × 1 vector bT such that bT a.s. −→ b, where b is k × 1, if g is continous at b, then g(bT) a.s. −→ g(b). This results is one of the most important in this Chapter, because consistency results for many of our estimators follows by simply applying this Proposition. 1.3 Convergence in Probability A weaker stochastic convergence concept is that of convergence in probability. Definition: Let {bT } be a sequence of real-valued random variables. If there exists a real num￾ber b such that for every δ > 0, such that Pr(s : |bT (s)−b| < δ) → 1, as T → ∞, then bT converge in probability to b, written as bT p −→ b or plim bT = b. Example: Let Z¯ T ≡ T −1 PT t=1 Zt , where {Zt} is a sequence of random variables such that E(Zt) = µ, V ar(Zt) = σ 2 < ∞ for all t and Cov(Zt , Zτ ) = 0 fort 6= τ . Then Z¯ T p −→ µ by the Chebyshev weak law of large numbers. See the plot of Hamilton p.184. When the plim of a sequence of estimator (such as {Z¯ T } ∞ T =1) is equal to the true population parameter (in thius case, µ), the estimator is said to be consistent. Convergence in probabbility is also referred as weak consistency, and since this has been the most familiar stochastic convergence concept in econometrics, the word ”weak” is often simply dropped. 4