time 2r ¤ 2n and then a return in 2n ’ 2r units of time, and the sum in (4.19) is

100 CHAPTER 4. SPACE AND TIME AVERAGES

over the possible times of ¬rst return. If we substitute (4.19) into the ¬rst sum

in (4.18) it becomes u2k while substituting (4.19) into the second term yields

u2n’2k . Thus (4.18) becomes

b2k,2n = u2k u2n’2k

which is our desired result.

Proof of Prop.4.4.3. This follows from Prop.4.4.1 because of the symmetry

of the whole picture under rotation through 180—¦ and a shift: The probability in

the lemma is the probability that S2k = S2n but Sj = S2n for j < 2k. Reading

the path rotated through 180—¦ about the end point, and with the endpoint

shifted to the origin, this is clearly the same as the probability that 2n ’ 2k is

the last visit to the origin. QED

Proof of Prop 4.4.4. The probability that the maximum is achieved at 0 is

the probability that S1 ¤ 0, . . . , S2n ¤ 0 which is u2n by (4.16). The probability

that the maximum is ¬rst obtained at the terminal point, is, after rotation and

translation, the same as the probability that S1 > 0, . . . , S2n > 0 which is 1 u2n

2

by (4.14). If the maximum occurs ¬rst at some time l in the middle, we combine

these results for the two portions of the path - before and after time - together

with (4.9) to complete the proof. QED

4.5 The Beta distributions.

The arc sine law is the special case a = b = 1 of the Beta distribution with

2

parameters a, b which has probability density proportional to

ta’1 (1 ’ t)b’1 .

So long as a > 0 and b > 0 the integral

1

ta’1 (1 ’ t)b’1 dt

B(a, b) =

0

converges, and was evaluated by Euler to be

“(a)“(b)

B(a, b) =

“(a + b)

where “ is Euler™s Gamma function. So the Beta distributions with A > 0, b > 0

are given by

1

ta’1 (1 ’ t)b’1 .

B(a, b)

1

We characterized the arc sine law (a = b = 2 ) as being the unique probability

density invariant under L4 . The case a = b = 0, where the integral does

not converge, also has an interesting characterization as an invariant density.

Consider transformations of the form

at + b

t’

ct + d

101

4.5. THE BETA DISTRIBUTIONS.

500

400

300

200

100

0

-100

0 2 4 6 8 10 12

4

x 10

Figure 4.2: A random walk with 100,000 steps. The last zero is at time 3783.

For the remaining 96,217 steps the path is positive. According to the arc sine

law, with probability 1/5, the particle will spend about 97.6 percent of its time

on one side of the origin.

102 CHAPTER 4. SPACE AND TIME AVERAGES

where the matrix

a b

c d

is invertible. Suppose we require that the transformation preserve the origin

and the point t = 1. Preserving the origin requires that b = 0, while preserving

the ™ point t = 1 requires that a = c + d. Since b = 0 we must have ad = 0 for

the matrix to be invertible. Since multi[lying all the entries of the matrix by

the same non-zero scalar does not change the transformation, we may as well

assume that d = 1, and hence the family transformations we are looking at are

at

t’

φa : , a = 0.

(a ’ 1)t + 1

Notice that

φa —¦ φb = φab .

Our claim is that, up to scalar multiple, the density

1

ρ(t) =

t(1 ’ t)

is the unique density such that the measure

ρ(t)dt

is invariant under all the transformations φa . Indeed,

a

φa (t) =

[1 ’ t + at]2

so the condition of invariance is

a

ρ(φa (t)) = ρ(t).

[1 ’ t + at]2

Let us normalize ρ by

1

ρ = 4.

2

Then

1 a s

”s= ”a=

s = φa .

1’s

2 1+a

1

So taking t = in the condition for invariance and a as above, we get

2

1 1s2 1