Weak convergence of random functions

In order to establish the limiting distribution of (29.14), and other asymptotic results, we need to extend the well known concept of convergence in distribution of random variables to convergence in distribution of a sequence of random functions. Recall that for random variables Xn, X, Xn ^ X in distribution if the distribution function Fn(x) of Xn converges pointwise to the distribution function F(x) of X in the continuity points of F(x). Moreover, recall that distribution func­tions are uniquely associated to probability measures on the Borel sets,5 i. e. there exists one and only one probability measure ц„(В) on the Borel sets B such that F„(x) = ц„((- <ж, x]), and similarly, F(x) is uniquely associated to a probability measure ц on the Borel sets, such that F(x) = ц((-^, x]). The statement Xn ^ X in distribution can now be expressed in terms of the probability measures ц n and ц : ц„(В) ^ ц(В) for all Borel sets B with boundary SB satisfying ц(5В) = 0.

In order to extend the latter to random functions, we need to define Borel sets of functions. For our purpose it suffices to define Borel sets of continuous func­tions on [0, 1]. Let C[0, 1] be the set of all continuous functions on the unit interval [0, 1]. Define the distance between two functions f and g in C[0, 1] by the sup-norm: p( f, g) = sup0<x£1 |f(x) – g(x)|. Endowed with this norm, the set C[0, 1] becomes a metric space, for which we can define open subsets, similarly to the concept of an open subset of R: A set B in C[0, 1] is open if for each function f in B we can find an є > 0 such that {g E C[0, 1] : p(g, f) < є} C B. Now the smallest o-algebra of subsets of C[0, 1] containing the collection of all open subsets of C[0, 1] is just the collection of Borel sets of functions in C[0, 1].

A random element of C[0, 1] is a random function W(x), say, on [0, 1], which is continuous with probability 1. For such a random element W, say, we can define a probability measure ц on the Borel sets B in C[0, 1] by ц(В) = P(W E B). Now a sequence W* of random elements of C[0, 1], with corresponding probability

measures pn, is said to converge weakly to a random element W of C[0, 1], with corresponding probability measure p, if for each Borel set B in C[0, 1] with boundary SB satisfying p(SB) = 0, we have pn(B) ^ p(B). This is usually denoted by: W* ^ W(on [0, 1]). Thus, weak convergence is the extension to random functions of the concept of convergence in distribution.

In order to verify that W* ^ W on [0, 1], we have to verify two conditions. See Billingsley (1963). First, we have to verify that the finite distributions of W* con­verge to the corresponding finite distributions of W, i. e. for arbitrary points x1,…, xm in [0, 1], (W*(x1), …, W*(xm)) ^ (W(x1), …, W(xm)) in distribution. Second, we have to verify that W* is tight. Tightness is the extension of the concept of stochastic boundedness to random functions: for each e in [0, 1] there exists a compact (Borel) set K in C[0, 1] such that pn(K) > 1 – e for n = 1, 2,… Since convergence in distribution implies stochastic boundedness, we cannot have convergence in dis­tribution without stochastic boundedness, and the same applies to weak conver­gence: tightness is a necessary condition for weak convergence.

As is well known, if Xn ^ X in distribution, and Ф is a continuous mapping from the support of X into a Euclidean space, then by Slutsky’s theorem, Ф(ХП) ^ Ф(Х) in distribution. A similar result holds for weak convergence, which is known as the continuous mapping theorem: if Ф is a continuous mapping from C[0, 1] into a Euclidean space, then W* ^ W implies Ф^П) ^ Ф^) in distribution. For example, the integral Ф( f) = ff(x)2dx with f Є C[0, 1] is a continuous mapping from C[0, 1] into the real line, hence W* ^ W implies that /W*(x)2dx ^ /W(x)2dx in distribution.

The random function Wn defined by (29.8) is a step function on [0, 1], and therefore not a random element of C[0, 1]. However, the steps involved can be smoothed by piecewise linear interpolation, yielding a random element Wn* of C[0, 1] such that sup0<x£11 W*(x) – Wn(x)| = op(1). The finite distributions of W* are therefore asymptotically the same as the finite distributions of Wn. In order to analyze the latter, redefine Wn as

1 [nx]

Wn(x) = ^ et for x Є [n-1, 1], Wn(x) = 0 for x Є [0, n-1), et is iid N(0, 1).

л/n t=1

(29.15)

(Thus, et = u/a), and let

Wn*(x) = Wn

t = 1, …, n, W*(0) = 0.

image698

t – 1 t

 

= Wn(x) + (nx – (t – 1)) – j= for x

4n

 

n n

 

image699

Then

max | et |

Подпись:Подпись:sup | W*(x) – Wn(x)l < ^

0<x<i n

Подпись: Wn(x) Wn( y) - Wn(x) image703 Подпись: N2 Подпись: ґ 0Л v0y image706

The latter conclusion is not too hard an exercise.6 It is easy to verify that for fixed 0 < x < y < 1 we have

Подпись: in distribution,Подпись: (29.18)f W(x) л

W( y) – W(x)y

Подпись: f W(x) л W( y) - W(x) image710 image711 Подпись: (29.19)

where W(x) is a random function on [0, 1] such that for 0 < x < y < 1,

This random function W(x) is called a standard Wiener process, or Brownian motion. Similarly, for arbitrary fixed x, y in [0, 1],

(29.20)

(W*(1),

W*(x)dx,

Wn*(x)2dx,

xW*(x)dx)T ^

Подпись: (W(1),
Подпись: W(x)dx,
Подпись: W(x)2dx,
Подпись: xW(x)dx)T
Подпись: (29.21)

and it follows from (29.17) that the same applies to W*. Therefore, the finite distributions of W* converge to the corresponding finite distributions of W. Also, it can be shown that W* is tight (see Billingsley, 1963). Hence, W* ^ W, and by the continuous mapping theorem,

in distribution. This result, together with (29.17), implies that:

Lemma 1. For Wn defined by (29.15), (Wn(1), /Wn(x)dx, /Wn(x)2dx, JxWn(x)dx)T converges jointly in distribution to (W(1), /W(x)dx, /W(x)2dx, JxW(x)dx)T.

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