Moments of the positive-part Brownian bridge area

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

Numbers

$n$

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

0.1566642671644375314009853303006903283129366712881211447893702235213933534129901234341622398986285791

1/20

0.02172492767319348579974601260029104162152051496378242437508844896488071893031698977309671686094263500

211/18480

0.006876919488770984574814806092852858853324621426433726428088548891612036762336047462757371620364198029

75233/16336320

0.003363726711788444768609464048800874916380655981079917274186329927468320563750814957504072184892612974

75516257/28555887360

0.002216275403080094493851059159739809784973862663721130261546872160039679508018671955934231627356727298

10:

292964136763/149061732019200

11:

0.001833517603342294109299157800715258541521973482296296455234191354783242043803844261519504596008434124

12:

1158609985088903/646927916963328000

Definition

Let $b$ be the centered Gaussian bridge on $[0,1]$ with covariance $\min(s,t)-st$, and write $x_+=\max(x,0)$ [1]. The table gives the raw moments $M_n=\mathbb E[A_{\rm br+}^n]$ of the positive-part area $A_{\rm br+}=\int_0^1 b(t)_+\,dt$.

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)

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

(3)

$M_n=\dfrac{\sqrt{\pi}\,2^{-n/2}n!}{\Gamma((3n+1)/2)}P_n$ for $n\geq0$, where $P_n$ is defined by (2) and $\Gamma$ is the Gamma function [1], equation (240).

(4)

$M_1=\tfrac18\sqrt{\pi/2}$, $M_2=\tfrac1{20}$, $M_3=\tfrac{71}{4096}\sqrt{\pi/2}$ and $M_4=\tfrac{211}{18480}$ [1], Table 6.

Comments

(5)

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

(6)

The area $A_{\rm br+}$ is denoted $\mathcal B_{\rm br}^{+}$ in [1], Section 24. The recurrence coefficient $P_n$ in (2) is Janson's $D_n^+$; it is an intermediate coefficient, while the entries are raw moments.

(7)

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

# 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)

Links

[2]

Brownian areas and their moments

Similar tables

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

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 bridge 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 $K_n$ and $P_n$ 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 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 with an exact quotient of Airy asymptotic series, using [1], Appendix C.2 and equations (104), (106), (242). 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 mean also follows by integrating the positive-part Gaussian marginal mean, and the first four moments agree with Table 6 of [1]. The attached checks test strict moment log-convexity with $M_0=1$, the recurrence on database readback, and the one-value program. See checks.py, derivation.md and numerical-evidence.json for methods, controls and measurements.