Lecture 2 Flashcards
Give the definition of an indicator function.
Derive the expectation of an indicator function.
State Markov’s inequality.
State and prove Markov’s inequality.
State the relation between the integrals of two nonnegative functions.
State the relation between Markov and Chebyshev inequality.
Define convergence in r-th mean.
What is mean square convergence?
State and prove the relation between convergence in probability and convergence in r-th mean.
State and prove the relation between convergence in r-th and s-th mean.
+ RHS converges, so LHS must converge. Convex function converges iff argument converges.
Define consistency in r-th mean.
State and prove the relation between r-th mean consistency and asymptotic unbiasedness.
+ I understand actually, it’s jensen with taking to the r-th power, and then taking to 1/r on both sides which shouldn’t affect since they’re both positive and convex.
Does r-th mean convergence have the slutzky type equivalent for continuous functions? If so, prove it. If not show a reason why not.
Derive the necessary conditions to show that:
State the relation between convergence in second mean and the covariance structure.
State and prove the relation between convergence in second mean and the covariance structure.
State Chebyshev’s Weak LLN and show how it relates to Theorem 7 (relationship between convergence in second mean and covariance matrix).
Does Chebyshev’s weak LLN apply under the case of homoskedasticity? Give an example of a case where the law does not apply.
Define linear processes.
Define Stationary Autoregressive of order p AR(p) and note the condition.
Define moving average of order q. MA(q).
Define autoregressive moving average of order p,q: ARMA(p,q)
Give the two results regarding the Autocovariance Structure for Linear processes.
Prove the first result regarding the Autocovariance Structure for Linear processes.
Prove the second result regarding the Autocovariance Structure for Linear processes.
