Krein condition

A condition in terms of the logarithmic normalized integral

\begin{equation} \tag{a1} K : = \int \frac { - \operatorname { ln } f ( . ) } { 1 + x ^ { 2 } } d x, \end{equation}

used to derive non-uniqueness or uniqueness of the moment problem for absolutely continuous probability distributions (cf. also Absolute continuity; Probability distribution). In (a1), $f$ is the density function of some distribution function $F$ having all moments $\alpha _ { k } = \int x ^ { k } d F ( x )$, $k = 1,2 , \dots$, finite, the integral is taken over the support of $F$ and the argument of $f ( . )$ is $x$ or $x ^ { 2 }$, depending on this support.

The general question of interest is: Does the moment sequence $\{ \alpha _ { k } : k = 1,2 , \ldots \}$ determine $F$ uniquely? If the answer is "yes" , one says that the moment problem has a unique solution, or that the distribution function $F$ is M-determinate. Otherwise, the moment problem has a non-unique solution, or that $F$ is M-indeterminate.

It is essential to note that the quantity $K$ defined in (a1) may "be equal to" $+ \infty$.

Hamburger moment problem.

In this problem, the support of $F$ is $( - \infty , \infty )$, the density $f ( x ) > 0$ for all $x \in ( - \infty , \infty )$ and all moments $\alpha _ { k } = \int _ { - \infty } ^ { \infty } x ^ { k } f ( x ) d x$, $k = 1,2 , \dots$, are finite. The values of $K$ belong to the interval $[ - 1 , + \infty ]$.

For this problem the following Krein conditions are used:

\begin{equation} \tag{a2} \int _ { - \infty } ^ { \infty } \frac { - \operatorname { ln } f ( x ) } { 1 + x ^ { 2 } } d x < \infty; \end{equation}

\begin{equation} \tag{a3} \int _ { - \infty } ^ { \infty } \frac { - \operatorname { ln } f ( x ) } { 1 + x ^ { 2 } } d x = \infty. \end{equation}

The following is true:

if (a2) holds, then $F$ is M-indeterminate, i.e. the moment problem has a non-unique solution;

if, in addition to (a3), the Lin condition below is satisfied, then $F$ is M-determinate, i.e. the moment problem has a unique solution.

Here, the following Lin condition is used: $f$ is symmetric and differentiable, and for some $x _ { 0 } > 0$ and $x \geq x_0$,

\begin{equation} \tag{a4} \frac { - x f ^ { \prime } ( x ) } { f ( x ) } \nearrow \infty , \quad x \rightarrow \infty. \end{equation}

Stieltjes moment problem.

In this problem, the support of $F$ is the real half-line $( 0 , \infty )$, the density $f ( x ) > 0$ for all $x \in ( 0 , \infty )$, and all moments $\alpha _ { k } = \int _ { 0 } ^ { \infty } x ^ { k } f ( x ) d x$, $k = 1,2 , \dots$, are finite. In this case the values of $K$ belong to the interval $[ - 1 / 2 , + \infty ]$.

In this case one uses the following Krein conditions

\begin{equation} \tag{a5} \int _ { 0 } ^ { \infty } \frac { - \operatorname { ln } f ( x ^ { 2 } ) } { 1 + x ^ { 2 } } d x < \infty; \end{equation}

\begin{equation} \tag{a6} \int _ { 0 } ^ { \infty } \frac { - \operatorname { ln } f ( x ^ { 2 } ) } { 1 + x ^ { 2 } } d x = \infty$. \end{equation} The following is true: if (a5) holds, then $F$ is M-indeterminate. if, in addition to (a6), the Lin condition below is satisfied, then $F$ is M-determinate. Here, the Lin condition is that $f$ be differentiable and that for some $x _ { 0 } > 0$ and $x \geq x_0$, \begin{equation} \tag{a7} \frac { - x f ^ { \prime } ( x ) } { f ( x ) } \nearrow \infty , \quad x \rightarrow \infty. \end{equation} From these four assertions one can derive several interesting results. In particular, one can easily show that the [[Log-normal distribution|log-normal distribution]] is M-indeterminate. This fact was discovered by Th.J. Stieltjes in 1894 (in other terms; see [[#References|[a1]]], [[#References|[a3]]]), and was later given in a probabilistic setting by others, see e.g. [[#References|[a4]]]. =='"`UNIQ--h-2--QINU`"'Examples in probability theory.== Suppose $X$ is a [[Random variable|random variable]] with a [[Normal distribution|normal distribution]]. Then: the distribution of $X ^ { 2 n + 1 }$ is M-indeterminate for all $n = 1,2 , \dots$; the distribution of $| X | ^ { r }$ is M-determinate for all $r \in ( 0,4 ]$; the distribution of $| X | ^ { r }$ is M-indeterminate for all $r > 4$. For details (direct constructions and using the Carleman criterion), see [[#References|[a2]]]. A proof of this result based on the Krein or Krein–Lin technique is given in [[#References|[a12]]]. Let $X$ be a random variable whose distribution is M-determinate. Using the Krein–Lin techniques, one can easily answer questions like: For which values of the real parameter $r$ does the distribution of the power $X ^ { r }$ and/or $| X | ^ { r }$ become M-indeterminate? Suppose the random variable $X$ has:a [[Normal distribution|normal distribution]]; an [[Exponential distribution|exponential distribution]]; a [[Gamma-distribution|gamma-distribution]]; a [[Logistic distribution|logistic distribution]]; or an inverse Gaussian distribution (cf. also [[Gauss law|Gauss law]]). Then in each of these cases the distribution of $X ^ { 2 }$ is M-determinate, while already $X ^ { 3 }$ has an M-indeterminate distribution, i.e. $3$ is the minimal integer power of $X$ destroying the determinacy of the distribution of $X$. For details see [[#References|[a12]]]. A more general problem is to describe classes of functions of random variables (not just powers) preserving or destroying the determinacy of the probability distributions of the given variables. =='"`UNIQ--h-3--QINU`"'Generalization.== There is a more general form of the Krein condition, which requires instead of (a2) that \begin{equation*} \int _ { - \infty } ^ { \infty } \left[ \frac { - \operatorname { ln } F _ { \text{ac} } ^ { \prime } ( x ) } { 1 + x ^ { 2 } } \right] d x < \infty , \end{equation*} where $F _ { \text{a.c.} }$ is the absolutely continuous part of the distribution function $F$, see [a8].

The Krein condition, in conjunction with the Lin condition, is used for absolutely continuous distributions whose densities are positive in both Hamburger and Stieltjes problems. [a7] contains an extension of the Krein condition for indeterminacy as well as a discrete analogue applicable to distributions concentrated on the integers.

The Krein condition can also be used for other purely analytic problems, see [a3] and [a9].

The book [a1] is the basic source describing the progress in the moment problem, providing also an intensive discussion on the Krein condition. For distributions on the real line, this condition was introduced by M.G. Krein in 1944, see [a5]. For recent (1998) developments involving the Krein condition see [a3], [a6], [a7], [a9], [a10]. Several applications of the Krein condition are given in [a11] and [a12].

References

[a1]	N.I. Akhiezer, "The classical moment problem" , Hafner (1965) (In Russian)
[a2]	C. Berg, "The cube of a normal distribution is indeterminate" Ann. of Probab. , 16 (1988) pp. 910–913
[a3]	C. Berg, "Indeterminate moment problems and the theory of entire functions" J. Comput. Appl. Math. , 65 (1995) pp. 27–55
[a4]	C.C. Heyde, "On a property of the lognormal distribution" J. R. Statist. Soc. Ser. B , 29 (1963) pp. 392–393
[a5]	M.G. Krein, "On one extrapolation problem of A.N. Kolmogorov" Dokl. Akad. Nauk SSSR , 46 : 8 (1944) pp. 339–342 (In Russian)
[a6]	G.D. Lin, "On the moment problem" Statist. Probab. Lett. , 35 (1997) pp. 85–90
[a7]	H.L. Pedersen, "On Krein's theorem for indeterminacy of the classical moment problem" J. Approx. Th. , 95 (1998) pp. 90–100
[a8]	Yu.V. Prohorov, Yu.A. Rozanov, "Probability theory" , Springer (1969) (In Russian)
[a9]	B. Simon, "The classical moment problem as a self-adjoint finite difference operator" Adv. Math. , 137 (1998) pp. 82–203
[a10]	E.V. Slud, "The moment problem for polynomial forms of normal random variables" Ann. of Probab. , 21 (1993) pp. 2200–2214
[a11]	J. Stoyanov, "Counterexamples in probability" , Wiley (1997) (Edition: Second)
[a12]	J. Stoyanov, "Krein condition in probabilistic moment problems" Bernoulli 6, No. 5, 939-949 (2000) Zbl 0971.60017