Moments of the Brownian excursion area

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Numbers

$n$

$\mathbb E[A_{\rm ex}^{n}]$

0.6266570686577501256039413212027613132517466851524845791574808940855734136519604937366489595945143165

5/12

0.2937455009333203713768474943137943655867562586652271464800691691026125376493564814390541998099285859

221/1008

0.1728814666951312602374154523825977255797055064019305601679573755656000872623816791802766905131350531

82825/576576

0.1254219342635216646449538726263868086571056725095126759330590473544683523926017528713843615724901243

256406305/2234808576

0.1096481081341021772048593752654193208197864890488459601616462378213039617523051236788471546205043406

10:

304702375/2790982656

11:

0.1129601194889031649478811749209572171553203016873852769372926853845101094284113124678814107236396715

12:

89585870372335/739346190815232

Definition

Let $b_1,b_2,b_3$ be independent standard Brownian bridges on $[0,1]$, each with covariance $\min(s,t)-st$, and let $e(t)=\sqrt{b_1(t)^2+b_2(t)^2+b_3(t)^2}$ be the normalized Brownian excursion [1]. The table gives the raw moments $M_n=\mathbb E[A_{\rm ex}^n]$ of its area $A_{\rm ex}=\int_0^1 e(t)\,dt$.

Parameters

$n$

— moment order ($n\geq 1$)

Formulas

(1)

$K_0=-\tfrac12$ and $K_n=\tfrac{3n-4}{4}K_{n-1}+\sum_{j=1}^{n-1}K_jK_{n-j}$ for $n\geq1$, with an empty sum equal to zero [1], equation (7).

(2)

$M_n=\dfrac{4\sqrt{\pi}\,2^{-n/2}n!}{\Gamma((3n-1)/2)}K_n$ for $n\geq1$, where $K_n$ is defined by (1) and $\Gamma$ is the Gamma function [1], equation (6).

Comments

(3)

The underlying Brownian motion has variance $t$ and generator $\tfrac12\,d^2/dx^2$, and the duration is one. For duration $T>0$, Brownian scaling gives $A_{{\rm ex},T}\overset{d}=T^{3/2}A_{\rm ex}$, so its $n$th raw moment is $T^{3n/2}M_n$ [1].

(4)

Flajolet and Louchard call the distribution of $2^{3/2}A_{\rm ex}$ the Airy distribution; their $n$th moment is $2^{3n/2}M_n$ [1], equation (2). The values here use the unscaled Brownian excursion area.

(5)

Every even-order moment is rational, and every odd-order moment is a rational multiple of $\sqrt{\pi/2}$, by (2) and the integer and half-integer values of the Gamma function. The normalization is $M_0=1$.

Programs

(P1)

Python

import numberdb.sage
from math import factorial
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

def moment(n):
    """Raw moment of the standard Brownian excursion area, n >= 1."""
    if n < 1:
        raise ValueError('n must be positive')
    k = [-QQ(1)/QQ(2)]
    for j in range(1, n+1):
        k.append(QQ(3*j-4)/QQ(4)*k[j-1]
                 + sum((k[i]*k[j-i] for i in range(1,j)), QQ(0)))
    m = n//2
    if n%2:
        r = RealBallField(512)
        q = QQ(4*factorial(n))*k[n]/QQ(2**m*factorial(3*m))
        return r(q)*(r.pi()/r(2)).sqrt()
    j = 3*m-1
    return QQ(4*factorial(n)*4**j*factorial(j))*k[n]/QQ(2**m*factorial(2*j))

# For example, the mean:
print(moment(1))

References

[1]

S. Janson, Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas, Probability Surveys 4 (2007), 80–145. (arXiv) (doi)

Links

[2]

Brownian excursion area and related Brownian areas

Similar tables

Zeros of the Airy function $\operatorname{Ai}$ — these zeros determine the spectral expansion of the excursion-area Laplace transform in [1], equation (80)

Values of the Gamma function at rational numbers — the Gamma values appearing in the moment formula

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the first twelve positive integer raw moments of the unit-duration standard Brownian excursion area; moment order indexes the coefficients of its Laplace transform, and twelve is an initial editorial truncation of this discrete sequence)

How they were obtained:

The recurrence for $K_n$ and the rational prefactors were evaluated exactly over $\mathbb Q$. Even moments are stored as exact rational numbers. Odd moments are formed from the rational prefactor and an Arb enclosure of $\sqrt{\pi/2}$ at 513 bits, then written to 100 significant digits; the lowest measured relative accuracy was 510 bits. Each written decimal denotes the interval within one unit of its last place.

An independent exact calculation used the formal logarithm of the factorial series for graph enumeration and Spencer’s moment relation, [1], equations (34), (36), and (63)–(64), for orders 2 through 12. The mean was independently recovered from the Maxwell radial mean and the beta integral over bridge time. The four positive moments in Table 1 of [1] agree. Direct Arb evaluation of the Gamma formula checks the cancellation at all twelve orders. The attached checks also test the written enclosures and strict log-convexity, with the zeroth moment equal to one. See checks.py and numerical-evidence.json for the controls and results.