Moments of the Brownian meander area

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Numbers

$n$

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

0.9399856029866251884059119818041419698776200277287268687362213411283601204779407406049734393917714748

59/60

1.138263816116616439085284040465953166648680502327755192610268030272623583391256365576335024263473270

5345/3696

1.998409217101052690241224006144478260225078758263024117440857475959626293152448651813982199376549126

29181239/9801792

4.743118101851495708656534431283069555886400276263460303407432413930916826254912090630551918470624272

412489581535/51400597248

14.33362687102359914440474651494517668380878038059399322423019942073406318809985136875721221357872060

10:

53446407830573/1987489760256

11:

52.76792856635463452662673847373621057385543365490724131248862804839130831075674135899635140506288750

12:

90075877415461967/834745699307520

Definition

Let $m$ be the weak limit on $C[0,1]$, as $\varepsilon\downarrow0$, of standard Brownian motion started at $\varepsilon>0$ and conditioned not to hit zero before time $1$, with its endpoint free [1]. The table gives the raw moments $M_n=\mathbb E[A_{\rm me}^{n}]$ of the area $A_{\rm me}=\int_0^1m(t)\,dt$.

Parameters

$n$

— moment order ($n\geq1$)

Formulas

(1)

The auxiliary coefficients satisfy $K_0=-\tfrac12$ 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)

The auxiliary coefficients satisfy $Q_0=1$ and $Q_n=\tfrac{3n-2}{2}Q_{n-1}+2\sum_{j=1}^{n}K_jQ_{n-j}$ for $n\geq1$, where $K_n$ is defined by (1) [1], equation (199).

(3)

$M_n=\dfrac{\sqrt{\pi}\,2^{-n/2}n!}{\Gamma((3n+1)/2)}Q_n$ for $n\geq0$, where $Q_n$ is defined by (2) and $\Gamma$ is the Gamma function [1], equation (194).

(4)

For the standard Brownian meander $m_T$ of duration $T>0$, let $A_{{\rm me},T}=\int_0^T m_T(t)\,dt$. Brownian scaling gives $A_{{\rm me},T}\overset{d}=T^{3/2}A_{\rm me}$ and $\mathbb E[A_{{\rm me},T}^{n}]=T^{3n/2}M_n$ [1], section 1 and Appendix C.3.

Comments

(5)

The underlying Brownian motion has zero drift, variance $t$, and generator $\tfrac12\,d^2/dx^2$. The meander has duration one and is not conditioned to return to zero.

(6)

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

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 meander 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 $K_n$ and $Q_n$ 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 killed-Brownian-motion Feynman-Kac resolvent equation, using the framework of [1], Appendix C.3. The independent computation evaluates Gamma factors directly in Arb at 768 bits. The zeroth moment is a control, and the first four positive moments agree with Table 4 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 methods, controls and measurements.