Abstract
We recall from Chap. 8 that discussion of random variables (r.v.) takes place in a probability space (Ω, \(\mathcal{A}\), \(\mathcal{P}\)), where Ω is the sample space, \(\mathcal{A}\) is the σ-algebra and \(\mathcal{P}\) is the probability measure.
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- 1.
Part of this chapter is an adaptation of a set of Lectures given in the Spring of 2005, at the University of Cyprus, whose purpose was stated as “The purpose of these Lectures is to set forth, in a convenient fashion, the essential results from probability theory necessary to understand classical econometrics.” All page references, unless otherwise indicated, are to: Dhrymes (1989).
- 2.
The notation a.c. means almost certainly; an alternative notation is a.s. which means almost surely. More generally, the notation, say, \(X_{n}\stackrel{\mathrm{a.c}}{\rightarrow }X_{0}\), or \(X_{n}\stackrel{\mathrm{P}}{\rightarrow }X_{0}\) means that the sequence of random variables {X n : n ≥ 1} converges to the random variable X 0, respectively, almost certainly or in probability. These concepts will be defined more precisely below.
- 3.
This is a summary of relevant results presented earlier in Chap. 8.
- 4.
The implication in item (3) is easily established using Proposition 8.12, Chap. 8, (Generalized Chebyshev Inequality) as follows: define \(Y _{n} = \vert X_{n} - X_{0}{\vert }^{p}\) and note that the Y-sequence consists of non-negative integrable rvs obeying EY n = s n , such that s n converges to zero with n.
- 5.
Strictly speaking, this and the preceding are implied by the following condition, which states that the sequence of rvs in question possesses finite (2 + δ)th moments, for arbitrary δ > 0. Moreover this last requirement implies the Lindeberg condition.
- 6.
The function I in the equation below is the indicator function, which assumes the value 1, if \(\vert X_{in}\vert \geq \frac{1} {r}\) and the value zero otherwise.
- 7.
This discussion owes a great deal to Billingsley (1995), Shiryayev (1984), and Stout (1974) which contain a very lucid description of the concepts and issues involved in ergodicity, including the role played by Kolmogorov’s extension theorem we discussed in Chap. 8 (Proposition 8.6), and Kolmogorov’s Zero-One Law, Proposition 8.25.
- 8.
This is a property referred to under certain circumstances as mixing.
- 9.
See Billingsley (1995), p. 325, who also gives a counterexample of an ergodic sequence which is not mixing.
- 10.
Actually if we wished we could invoke the earlier discussion on ergodicity to argue that the convergence is actually with probability one, which of course implies convergence in probability.
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Dhrymes, P.J. (2013). LLN, CLT and Ergodicity. In: Mathematics for Econometrics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8145-4_9
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