4.3 The mean ergodic theorem

The purpose of this section is to prove

Theorem 4.3.1 von Neumann™s mean ergodic theorem. Let U : H ’ H be an

isometry of a Hilbert space, H. Then for any f ∈ H , the limit

1 ˆ

U kf = f

lim (4.6)

n

ˆ

exists in the Hilbert space sense, and the limiting element f is invariant, i.e.

ˆˆ

U f = f.

Proof. The limit, if it exists, is invariant as we have seen. If U were a unitary

operator on a ¬nite dimensional Hilbert space, H, then we could diagonalize U ,

and hence reduce the theorem to the one dimensional case. A unitary operator

on a one dimensional space is just multiplication by a complex number of the

form ei± . If ei± = 1, then

1 1 ’ ein±

1

(1 + ei± + · · · + e(n’1)i± ) = ’ 0.

n 1 ’ ei±

n

On the other hand, if ei± = 1, the expression on the left is identically one.

This proves the theorem for ¬nite dimensional unitary operators. For an in¬-

nite dimensional Hilbert space, we could apply the spectral theorem of Stone

(discovered shortly before the proof of the ergodic theorem) and this was von

Neumann™s original method of proof.

Actually, we can proceed as follows:

Lemma 4.3.1 The orthogonal complement of the set, D, of all elements of the

form U g ’ g, consists of invariant elements.

Proof. If f is orthogonal to all elements in D, then,in particular, f is orthogonal

to U f ’ f , so

0 = (f, U f ’ f )

and

(U f, U f ’ f ) = (U f, U f ) ’ (U f, f ) = (f, f ) ’ (U f, f )

since U is an isometry. So

(U f, U f ’ f ) = (f ’ U f, f ) = 0.

So

(U f ’ f, U f ’ f ) = (U f, U f ’ f ) ’ (f, U f ’ f ) = 0,

or

Uf ’ f = 0

which says that f is invariant.

So what we have shown, in fact, is

93

4.4. THE ARC SINE LAW

Lemma 4.3.2 The union of the set D with the set, I, of the invariant functions

is dense in H.

In fact, if f is orthogonal to D, then it must be invariant, and if it is orthogonal

to all invariant functions it must be orthogonal to itself, and so must be zero.

So (D ∪ I)⊥ = 0, so D ∪ I is dense in H.

Now if f is invariant, then clearly the limit(4.6) exists and equals f . If

f = U g ’ g, then the expression on the left in (4.6) telescopes into

1n

(U g ’ g)

n

which clearly tends to zero. Hence, as a corollary we obtain

Lemma 4.3.3 The set of elements for which the limit in (4.6) exists is dense

in H.

Hence the mean ergodic theorem will be proved, once we prove

Lemma 4.3.4 The set of elements for which the limit in (4.6) exists is closed.

Proof. If

1 1

U k gi ’ g i , U k gj ’ g j ,

ˆ ˆ

n n

and

gi ’ gj < ,

then

1 1

U k gi ’ U k gj < ,

n n

so

gi ’ gj < .

ˆ ˆ

So if {gi } is a sequence of elements converging to f , we conclude that {ˆi }

g

ˆ

converges to some element, call it f. If we choose i su¬ciently large so that

gi ’ f < , then

1 1 1

ˆ ˆˆ

U kf ’f ¤ U k (f ’gi ) U k gi ’ˆi gi ’ f ¤ 3 ,

+ g +

n n n

proving the lemma and hence proving the mean ergodic theorem.

4.4 the arc sine law

The probability distribution with density

1

σ(x) =

x(1 ’ x)

π

94 CHAPTER 4. SPACE AND TIME AVERAGES

is called the arc sine law in probability theory because, if I is the interval

I = [0, u] then

u √

1 2

Prob x ∈ I = Prob 0 ¤ x ¤ u = = arcsin u. (4.7)

π

x(1 ’ x)

π

0

We have already veri¬ed this integration because I = h(J) where

πt

h(t) = sin2 , J = [0, v], h(v) = u,

2

and the probability measure we are studying is the push forward of the uniform

distribution. So

Prob h(t) ∈ I = Prob t ∈ J = v.

The arc sine law plays a crucial role in the theory of ¬‚uctuations in random

walks. As a cultural diversion we explain some of the key ideas, following the

treatment in Feller [?] very closely.

Suppose that there is an ideal coin tossing game in which each player wins

or loses a unit amount with (independent) probability 1 at each throw. Let

2

S0 = 0, S1 , S2 , . . . denote the successive cumulative gains (or losses) of the ¬rst

player. We can think of the values of these cumulative gains as being marked

o¬ on a vertical s-axis, and representing the position of a particle which moves

up or down with probability 1 at each (discrete) time unit . Let

2

±2k,2n

denote the probability that up to and including time 2n, the last visit to the

origin occurred at time 2k. Let

2ν

2’2ν .

u2ν = (4.8)

ν

So u2ν represents the probability that exactly ν out of the ¬rst 2ν steps were in

the positive direction, and the rest in the negative direction. In other words, u2ν

is the probability that the particle has returned to the origin at time 2ν. We can

¬nd a simple approximation to u2ν using Stirling™s formula for an approximation

to the factorial: √ 1