Moments of the absolute Brownian motion area

edit · history · discussion · files · long url · integral probability theory

Numbers

$n$

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

0.5319230405352435705865947465791758246344781748865795435545677728942105678657357846683364452097425819

3/8

0.3330851420494501406292248055960077187592089523694533808448841054075651889254488365899344883099102358

903/2560

0.4269488344296140629064372227277551410607800199184325957545943601374364368588992908000776240149411103

37623/65536

0.8413759825276237479563841408300769748998116950431157864986964990876023280002075822573861351613277127

114752519/86507520

2.228265880777696729480657872076071817688766333320903211338690284200831591723300081610778049324130770

10:

189970427903/47982837760

11:

7.397951532331789401182768848425215883399265954450698254870974601654267057473376449201045767270245632

12:

274700964958021/18983755448320

Definition

Let $B$ be standard Brownian motion on $[0,1]$, started at zero, with mean zero and covariance $\min(s,t)$ [1]. The table gives the raw moments $M_n=\mathbb E[A_{\rm bm}^{n}]$ of its absolute area $A_{\rm bm}=\int_0^1|B(t)|\,dt$.

Parameters

$n$

— moment order ($n\geq1$)

Formulas

(1)

The auxiliary coefficients satisfy $D_0=1$ 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)

The auxiliary coefficients satisfy $L_0=1$ and $L_n=\tfrac{3n-1}{2}L_{n-1}-\sum_{j=1}^{n-1}D_{n-j}L_j$ for $n\geq1$, where $D_n$ is defined by (1) and an empty sum is zero [1], equation (190).

(3)

$M_n=\dfrac{2^{-n/2}n!}{\Gamma((3n+2)/2)}L_n$ for $n\geq0$, where $L_n$ is defined by (2) and $\Gamma$ is the Gamma function [1], equation (181).

(4)

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

Comments

(5)

The process has variance $t$ and generator $\tfrac12\,d^2/dx^2$. The duration is one, and the absolute value is taken before integration.

(6)

Every even-order moment is rational, and every odd-order moment is a rational multiple of $\sqrt{2/\pi}$, by (3) 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 Brownian absolute area."""
    if n < 1:
        raise ValueError('n must be positive')
    d, ell = [QQ(1)], [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))
        ell.append(QQ(3*j-1)/QQ(2)*ell[j-1]
                   - sum((d[j-k]*ell[k] for k in range(1,j)), QQ(0)))
    m = n//2
    if n%2 == 0:
        return QQ(factorial(n))*ell[n]/QQ(2**m*factorial(3*m))
    k = 3*m+2
    q = QQ(factorial(n)*4**k*factorial(k))*ell[n]/QQ(2**(m+1)*factorial(2*k))
    r = RealBallField(513)
    return r(q)*(r(2)/r.pi()).sqrt()

# 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

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 motion 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 recurrences for $D_n$ and $L_n$ 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 $\sqrt{2/\pi}$ 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, using the framework of [1], Appendix C.1. The independent computation evaluates Gamma factors directly in Arb at 768 bits. The mean follows separately by integrating the folded-normal marginal mean, and the second moment follows from the joint-normal absolute product integral. The first four positive moments agree with Table 3 of [1]. The attached checks test the actual written intervals, the recurrence on stored values, and strict log-convexity including $M_0=1$. See checks.py, derivation.md and numerical-evidence.json for the methods, controls and measurements.