Moments of the absolute Brownian bridge area

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Numbers

$n$

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

0.3133285343288750628019706606013806566258733425762422895787404470427867068259802468683244797972571583

7/60

0.05140546266333106499094831150491401397768234526641475063401210459295719408863738425183448496673750253

19/720

0.01545223728867987370263624839879855777305332402353538635129530524966868036592968990903358030250145165

70753/7001280

0.007250089015820064917017644888681340643249947089549250528872497129983263983985554692983340230009267121

206530429/36714712320

0.004668916525464758066447476371868728301485700733632331152248799373797706146670277309070233103958507243

10:

1256447927/305663155200

11:

0.003813678010375084714755916948502299248277432659792747145229124222095837540843711500869493178456493594

12:

48970038774497/13202610550272000

Definition

Let $b$ be the centered Gaussian bridge on $[0,1]$ with covariance $\min(s,t)-st$ [1]. The table gives the raw moments $M_n=\mathbb E[A_{\rm br}^{n}]$ of its absolute area $A_{\rm br}=\int_0^1|b(t)|\,dt$.

Parameters

$n$

— moment order ($n\geq1$)

Formulas

(1)

$D_0=1$ and $D_n=\tfrac{3n-2}{4}D_{n-1}-\tfrac12\sum_{j=1}^{n-1}D_jD_{n-j}$ for $n\geq1$, with an empty sum equal to zero [1], equation (149).

(2)

$M_n=\dfrac{\sqrt{\pi}\,2^{-n/2}n!}{\Gamma((3n+1)/2)}D_n$ for $n\geq0$, where $D_n$ is defined by (1) and $\Gamma$ is the Gamma function [1], equation (135).

(3)

For a standard bridge $b_T(t)=B(t)-(t/T)B(T)$ of duration $T>0$, let $A_{{\rm br},T}=\int_0^T|b_T(t)|\,dt$. Brownian scaling gives $A_{{\rm br},T}\overset{d}=T^{3/2}A_{\rm br}$ and $\mathbb E[A_{{\rm br},T}^n]=T^{3n/2}M_n$ [1].

Comments

(4)

The bridge is $b(t)=B(t)-tB(1)$, where $B$ is Brownian motion started at zero with mean zero, variance $t$ and generator $\tfrac12\,d^2/dx^2$. The duration is one, and the absolute value is taken before integration.

(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 zeroth moment 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 unit-duration standard bridge absolute area."""
    if n < 1:
        raise ValueError('n must be positive')
    d = [QQ(1)]
    for j in range(1, n+1):
        d.append(QQ(3*j-2)/QQ(4)*d[j-1]
                 - sum((d[k]*d[j-k] for k in range(1,j)), QQ(0))/QQ(2))
    m = n//2
    if n%2:
        r = RealBallField(513)
        q = QQ(factorial(n))*d[n]/QQ(2**m*factorial(3*m+1))
        return r(q)*(r.pi()/r(2)).sqrt()
    return QQ(factorial(n)*4**(3*m)*factorial(3*m))*d[n]/QQ(2**m*factorial(6*m))

# 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 areas and their moments

Similar tables

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

Values of the Airy derivative $\operatorname{Ai}'$ — this derivative appears in the double Laplace transform of the absolute bridge area in [1], equation (133)

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 bridge absolute 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 $D_n$ and all rational prefactors were evaluated exactly over $\mathbb Q$. Even moments are stored as exact rational numbers. Odd moments use an Arb enclosure of $\sqrt{\pi/2}$ at 513 bits, with a measured minimum relative accuracy of 509 bits, and are written to 100 significant digits. Each decimal denotes an interval within one unit of its last place.

All twelve orders were compared with an independent polynomial solution of the Feynman-Kac resolvent equation and with an exact quotient of Airy asymptotic series [1], Appendix C.2 and equations (104)–(107), (137). The mean was independently obtained from the folded-normal marginal mean and a beta integral. The first four positive moments agree with Table 2 of [1]. The attached checks test the actual written intervals, the recurrence on the stored values, and strict log-convexity including $M_0=1$. See checks.py and numerical-evidence.json for methods, controls and measurements.