Hausdorff moment problem

Hausdorff moment problem

In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence { mn : n = 0, 1, 2, ... } be the sequence of moments

m_n  = \int_0^1 x^n\,d\mu(x)\,

of some Borel measure μ supported on the closed unit interval [0, 1]. In the case m0 = 1, this is equivalent to the existence of a random variable X supported on [0, 1], such that E Xn = mn.

The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a half-line [0, ∞), and in the Hamburger moment problem one considers the whole line (−∞, ∞).

In 1921, Hausdorff showed that { mn : n = 0, 1, 2, ... } is such a moment sequence if and only if the sequence is completely monotonic, i.e., its difference sequences satisfy the equation

(-1)^k(\Delta^k m)_n \geq 0

for all n,k ≤ 0. Here, Δ is the difference operator given by

m)n = mn + 1mn.

The necessity of this condition is easily seen by the identity

(-1)^k(\Delta^k m)_n = \int_0^1 x^n (1-x)^k d\mu(x),

which is ≥ 0, being the integral of an almost sure non-negative function. For example, it is necessary to have

\Delta^4 m_6 = m_6 - 4m_7 + 6m_8 - 4m_9 + m_{10} = \int x^6 (1-x)^4 d\mu(x) \geq 0.

See also

  • Total monotonicity

References

  • Hausdorff, F. "Summationsmethoden und Momentfolgen. I." Mathematische Zeitschrift 9, 74-109, 1921.
  • Hausdorff, F. "Summationsmethoden und Momentfolgen. II." Mathematische Zeitschrift 9, 280-299, 1921.
  • Feller, W. "An Introduction to Probability Theory and Its Applications", volulme II, John Wiley & Sons, 1971.
  • Shohat, J.A.; Tamarkin, J. D. The Problem of Moments, American mathematical society, New York, 1943.

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Moment problem — In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequences of moments More generally, one may consider for an arbitrary sequence of functions Mn. Contents 1 …   Wikipedia

  • Hamburger moment problem — In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence { αn : n = 1, 2, 3, ... }, does there exist a positive Borel measure μ on the real line such that:alpha n = int {… …   Wikipedia

  • Stieltjes moment problem — In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions that a sequence { mu; n , : n = 0, 1, 2, ... } be of the form:mu n=int 0^infty x^n,dF(x),for some nondecreasing function …   Wikipedia

  • Moment (mathematics) — Second moment redirects here. For the technique in probability theory, see Second moment method. See also: Moment (physics) Increasing each of the first four moments in turn while keeping the others constant, for a discrete uniform distribution… …   Wikipedia

  • Moment (mathématiques) — Pour les articles homonymes, voir Moment. En probabilités (mathématiques, statistiques), on définit le moment d ordre n>0 d une variable aléatoire X, s il existe, le nombre . Sommaire 1 …   Wikipédia en Français

  • Felix Hausdorff — Infobox Scientist name = Felix Hausdorff |300px image width = 300px caption = birth date = Birth date|1868|11|8 birth place = Breslau, Germany death date = death date and age|1942|1|26|1868|11|8 death place = Bonn, Germany residence = nationality …   Wikipedia

  • List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Function (mathematics) — f(x) redirects here. For the band, see f(x) (band). Graph of example function, In mathematics, a function associates one quantity, the a …   Wikipedia

  • Michael Atiyah — Sir Michael Atiyah Born 22 April 1929 (1929 04 22) (age 82) …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”