A Limit Theorem for Fluctuations

Abstract

To motivate the results of this chapter, consider the classical strong law of large numbers: Let {X _n } be i.i.d. random variables with E[X _n ] = µ, E[X ² _n ] < ∞. Let

$${S_0} = 0,\,{S_n}\mathop = \limits^{def} \frac{{\sum {_{i = 1}^n{X_i}} }}{n},\,n \geqslant 1.$$

. The strong law of large numbers (see, e.g., Section 4.2 of Borkar, 1995) states that

$$\frac{{{S_n}}} {n} \to \mu ,\;\;a.s.$$

. To cast this as a ‘stochastic approximation’ result, note that some simple algebraic manipulation leads to

$$\begin{array}{*{20}{l}} {{S_{n + 1}} = {S_n} + \frac{1}{{n + 1}}({X_{n + 1}} - {S_n})} \\ {\quad \quad = {S_n} + \frac{1}{{n + 1}}([\mu - {S_n}] + [{X_{n + 1}} - \mu ])} \\ {\quad \quad = {S_n} + a(n)(h({S_n}) + {M_{n + 1}})} \end{array}$$

for

$$a\left( n \right)\mathop = \limits^{def} \frac{1} {{n + 1}},\;\;h\left( x \right)\mathop = \limits^{def} \mu - x\;\forall x,\quad {M_{n + 1}}\mathop = \limits^{def} {X_{n + 1}} - \mu$$

. In particular, {a(n)} and {M_n+1} are easily seen to satisfy the conditions stipulated for the stepsizes and martingale difference noise resp. in Chapter 2. Thus this is a valid stochastic approximation iteration. Its o.d.e. limit then is

$$\dot x\left( t \right) = \mu - x\left( t \right),\;\;t \geqslant 0$$

which has µ as the unique globally asymptotically stable equilibrium. Its ‘scaled limit’ as in assumption (A5) of Chapter 3 is

$$\dot x\left( t \right) = - x\left( t \right),\;\;t \geqslant 0$$

which has the origin as the unique globally asymptotically stable equilibrium.