Rate of Convergence
This chapter completes the review of classical probabilistic techniques begun in Chapter 1. Here, we supplement the laws of large numbers, the consistency properties of estimators and the convergence properties of algorithms with central limit theorems an
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This chapter completes the review of classical probabilistic techniques begun in Chapter 1. Here, we supplement the laws of large numbers, the consistency properties of estimators and the convergence properties of algorithms with centrallimit theorems and results relating to the convergence in distribution of estimators or algorithms. Following the description of the main tools, we define the rate of convergence of the Robbins-Momo algorithm introduced in Chapter 1. Then we consider the simplest linear model, the autoregressive model. This is included here since the approach uses the tools of Section 2.1 and, moreover, since the autoregressive model represents the crossroads between Parts II, III and IV of this book. 2.1 Convergence in Distribution 2.1.1 Weak Convergence on aMetrie Space 2.1.2 Convergence in Distribution of Random Vectors 2.1.3 Central Limit Theorem for Martingales 2.1.4 Lindeberg' s Condition 2.1.5 Applications 2.2 Rate of Convergence of the Robbins-Monro Algorithm 2.2.1 Convergence in Distribution of the Robbins-Monro Algorithm 2.2.2 Rate of Convergence of Newton's estimator 2.3 Autoregressive Models 2.3.1 Spectral Radius 2.3.2 Stability 2.3.3 Random Geometrie Series 2.3.4 Explosive Autoregressive Model 2.3.5 Jordan Decomposition
2.1 Convergence in Distribution 2.1.1 Weak Convergenee on aMetrie Spaee Suppose that E is ametrie space with its Borel a-field, C b is the set of real-valued continuous bounded functions on E, P(E) is the set of probabilities defined on E and Ll is a distance on E.
Definition 2.1.1 1. A sequence (IIn ) of measures bounded on E converges weakly to 11 in Cb, (IIn ( for CI. Study the asymptotic behaviour of:
y'n(9n - (), Cn,2 - C2)' Deduce the asymptotic behaviour of
vn(Cn,l - CI, Cn,2 - C2).
o
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2. Rate of Convergence
Example 3 (r-adic Random Numbers). Suppose that (Xn ) is a sequence of independent, uniformly distributed random variables with values 0,1, ... r - 1, for some integer r. We suppose further that P(Xn =i) =Pi (0 ::::; i ::::; r - 1) and that 0< Pr-I< 1. The series L::I Xnr- n eonverges everywhere to a random variable X whieh has the r-adie deeomposition
X
=0, X I X 2 .•• X n ....
Suppose that p, denotes the distribution of X and show that p, is non-atomic, in other words that it is zero at all points. First we note that the probability of obtaining only (r - l)'s from a eertain position onwards is zero sinee for all m: P(Xn = r - 1; m
m) = O.
Let x E [0,1[; then there exists an r-adie decomposition, whieh is unique if we exc1ude deeompositions eontaining only (r - l)'s after a eertain position: for x =",00 wn=l xnr -n ,we have P(X = x) = p,(x) = PX 1PX2
•• •
Px n
•••
= O.
Thus, the distribution funetion F of X is eontinuous and for all n and k, 0 ::::; k < r, we have
o Special Cases. a) If all the Pi are equal to l/r, we see that F(x) =x for all x of the form k/r n , whenee for all x. Thus, p, is the Lebesgue measure on the interval [0, 1]. By virtue of the law of
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