1. (b) Converge in Lp)converge in Lq)converge in probability ) converge weakly, p q 1. Section 8: Asymptotic Properties of the MLE In this part of the course, we will consider the asymptotic properties of the maximum likelihood estimator. (c) Convergence in KL divergence )Convergence in total variation)strong convergence of measure )weak convergence, where i. n TV! X Y Be careful because this property does not hold for convergence in distributions, i.e. Convergence in probability of a sequence of random variables. For a set of random variables X n and a corresponding set of constants a n (both indexed by n, which need not be discrete), the notation = means that the set of values X n /a n converges to zero in probability as n approaches an appropriate limit. Relationships between convergence: (a) Converge a.c. )converge in probability )weak convergence. Y , X n Yn p! Equivalently, X n = o p (a n) can be written as X n /a n = o p (1), where X n = o p (1) is defined as, There are several useful properties of the sample mean and variance, we use later in the course, when the population distribution is normal. This is typically possible when a large number of random eﬀects cancel each other out, so some limit is involved. Theorem~\ref{thm-helly} can be thought of as a kind of compactness property for probability distributions, except that the subsequential limit guaranteed to exist by the theorem is not a distribution function. There is another version of the law of large numbers that is called the strong law of large numbers (SLLN). A sequence of random variables {Xn} is said to converge in probability to X if, for any ε>0 (with ε sufficiently small): Or, alternatively: To say that Xn converges in probability to X, we write: This property is meaningful when we have to evaluate the performance, or consistency, of an estimator of some parameters. However, our next theorem gives an important converse to part (c) in (7) , when the limiting variable is a constant. As we have discussed in the lecture entitled Sequences of random variables and their convergence, different concepts of convergence are based on different ways of measuring the distance between two random variables (how "close to each other" two random variables are).. We will discuss SLLN in Section 7.2.7. This section discusses three such deﬁnitions, or modes, of convergence; Section 3.1 presents a fourth. The general situation, then, is the following: given a sequence of random variables, In particular, we will study issues of consistency, asymptotic normality, and eﬃciency.Manyofthe proofs will be rigorous, to display more generally useful techniques also for later chapters. Convergence in probability is stronger than convergence in distribution: (iv) is one-way. Convergence a.s. makes an assertion about the distribution of entire random sequences of Xt’s. One has to think of all the Xt’s and Z To ensure that we get a distribution function, it turns out that a certain property … It follows that convergence with probability 1, convergence in probability, and convergence in mean all imply convergence in distribution, so the latter mode of convergence is indeed the weakest. Theorem 2 (Sample Mean and Variance of Normal Random Variables) Let X 1,X ... while the common notation for convergence in probability is X n It is called the "weak" law because it refers to convergence in probability. converges in probability to $\mu$. Convergence in Probability. 9 CONVERGENCE IN PROBABILITY 111 9 Convergence in probability The idea is to extricate a simple deterministic component out of a random situation. Also, as was emphasized in lecture, these convergence notions make assertions about different types of objects. Theorem for convergence in probability, and comes from the fact that you can apply the CMT to any function of X n and Y n. This is true because of the following property holds for convergence in probability: X n p! Definitions Small O: convergence in probability. 2.1.1 Convergence in Probability Sub sub sequences and a relation between convergence in probability and a.s convergence 1 Elementary proof for convergence in probability for square of random variable X and Y n p! ← 2.1 Modes of Convergence Whereas the limit of a constant sequence is unequivocally expressed by Deﬁnition 1.30, in the case of random variables there are several ways to deﬁne the convergence of a sequence.

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