Moments of the positive-part Brownian motion area

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Numbers

$n$

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

0.2659615202676217852932973732895879123172390874432897717772838864471052839328678923341682226048712910

17/96

0.1589436704456501621633753349897299190276833594006326969430910845195795863503567642282767235091016525

6989/40960

0.2084239824185162781044506184528423555121114355464778365933812248157557879212006305614254444103867105

37235311/132120576

0.4145156599617482578354237131971702584052116747910439449250405008069867800882679883541499656080408875

522258818027/797253304320

1.102251713072378502970413578634661502469454973463114827614535220190449006917745045608974893926104004

10:

1011667945773427/515911471595520

11:

3.667775058773663242571601081208048100344826637276661095917897340999241181202664119737018370070564292

12:

17585994035619348337/2449360062964039680

Definition

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

Parameters

$n$

— moment order ($n\geq1$)

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)

$Q_0=1$ 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)

$P_0=1$ and $P_n=\sum_{j=1}^{n}K_jP_{n-j}$ for $n\geq1$, where $K_n$ is defined by (1) [1], equation (245).

(4)

$L_n^+=\tfrac12\left(P_n+\sum_{j=0}^{n}P_jQ_{n-j}\right)$ for $n\geq0$, where $P_n$ and $Q_n$ are defined by (3) and (2) [1], equation (298).

(5)

$M_n=\dfrac{2^{-n/2}n!}{\Gamma((3n+2)/2)}L_n^+$ for $n\geq0$, where $L_n^+$ is defined by (4) and $\Gamma$ is the Gamma function [1], equation (266).

(6)

$M_1=\tfrac13\sqrt{2/\pi}$, $M_2=\tfrac{17}{96}$, $M_3=\tfrac{251}{1260}\sqrt{2/\pi}$ and $M_4=\tfrac{6989}{40960}$ [1], Table 8.

Comments

(7)

The process has variance $t$ and generator $\tfrac12\,d^2/dx^2$. Its endpoint is unconditioned, and the positive part is taken before integration.

(8)

The area $A_{\rm bm+}$ is denoted $\mathcal B_{\rm bm}^{+}$ in [1], Section 25, and $A^+$ in [2]. The coefficients $K_n$, $Q_n$, $P_n$ and $L_n^+$ in the recurrences are intermediates; the entries are raw moments. The notation $P_n$ stands for Janson’s $D_n^+$.

(9)

Every even-order moment is rational, and every odd-order moment is a rational multiple of $\sqrt{2/\pi}$, by (5) and the integer and half-integer values of the Gamma function. The zeroth moment is $M_0=1$.

(10)

These raw moments enter the Laguerre expansions of the density and distribution function in [2], Section 6.

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 motion positive-part area."""
    if n < 1 or int(n) != n:
        raise ValueError('n must be a positive integer')
    n = int(n)
    alpha = [QQ(factorial(6*j))/(QQ(576)**j*factorial(3*j)*factorial(2*j))
             for j in range(n+1)]
    beta, ell = [QQ(1)], [QQ(1)]
    for j in range(1,n+1):
        beta.append(alpha[j]+QQ(3*(2*j-1))/4*beta[j-1])
        ell.append((alpha[j]+beta[j])/QQ(2)
                   + sum(alpha[k]*ell[j-k]/QQ(6*k-1) for k in range(1,j+1)))
    if n%2 == 0:
        return QQ(factorial(n))*ell[n]/(QQ(2)**(n//2)*factorial(3*n//2))
    field = RealBallField(513)
    return (field(factorial(n))*field(ell[n])
            / (field(2)**(QQ(n)/2)*field(QQ(3*n+2)/2).gamma()))

# For example, moment(1) gives the mean.

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)

[2]

M. Perman and J. A. Wellner, On the distribution of Brownian areas, Annals of Applied Probability 6 (1996), 1091–1111. (doi)

Links

[3]

Brownian areas and their moments

[4]

On the distribution of Brownian areas, author-hosted full text

Similar tables

Moments of the absolute Brownian motion area — raw moments of $\int_0^1|B(t)|\,dt$ for the same standard Brownian motion

Moments of the positive-part Brownian bridge area — raw moments of the positive-part area when the standard Brownian path is conditioned to return to zero at time one

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 every raw moment of positive integer order $1\leq n\leq12$ of the unit-duration standard Brownian motion positive-part area; moment order indexes the coefficients of its Laplace transform, and twelve is an editorial truncation of this discrete sequence)

How they were obtained:

The recurrences for $K_n$, $Q_n$ and $P_n$, the convolution for $L_n^+$, and the 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 the interval within one unit of its last place.

All twelve orders were compared with an independent polynomial solution of the Brownian Feynman–Kac resolvent equation and an exact quotient of Airy asymptotic series, using [1], Appendix C.1 and equations (71), (184), and (268). Direct Arb Gamma evaluation at 768 bits checks the half-integer simplification and containment within each written decimal interval. The zeroth moment and the first resolvent polynomial serve as controls. The first two moments also follow from Gaussian marginal and joint-product integrals. Orders one through four agree with Table 8 of [1], and orders one through ten agree with the printed Table 1 of [2]. The attached checks test strict moment log-convexity with $M_0=1$, the identities on database readback, and the one-value program. See checks.py, derivation.md and numerical-evidence.json for methods, controls and measurements.