Копаясь еще в архивах
Apr. 19th, 2010 12:02 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
posicДополнение к гипотезе о рациональности/мероморфности ряда Гильберта кошулевой алгебры/стационарной 1-зависимой последовательности. Из писем к Б.Ц., осень 1994 года (предположительно).
Let $\Xi=(\xi_i : i\in Z)$ be a 1-dependent stationary stochastic
sequence of 0's and 1's, and let
$$a_n=P\{\xi_1=...=\xi_{n-1}=1\}, \qquad a_1=a_0=1.$$
Consider the power series $h(z)=\sum a_iz^i$; obviously, it converges
(at least) when $|z|<1$. We have found that it is {\it meromorphic}
for $|z|<2$. Moreover, the series $(1+h(z))^{-1}$ converges in this double
size circle. Indeed, the coefficients of the last series are just the
probabilities like $P\{\sum_{i=1}^n \xi_i\ {\rm is\ even}\}$ with a
multiplier like $2^{-n}$.
Conjecture. For any 1-dependent stationary stochastic sequence $\Xi$,
the series $h(z)$ is meromorphic on the whole complex plane. Moreover,
the sum [of the inverse squares of absolute values of the coordinates
of the poles] $\sum_{z:h(z)=\infty}|z|^{-2}$ (counting multiplicities)
is bounded by a constant.
For the block factor sequences, it is true---as well as I understand---%
and it is written in the book by Danford and Schvarz, the chapter
concerning Hilbert-Shmidt operators (Carleman inequalities).
Furthermore, there is a Tikhonov topology on the set of all $\Xi$:
$\Xi^{(s)}\rightarrow\Xi$ if the corresponding probabilities
$a_n^{(s)}\rightarrow a_n$ for any $n$. Since the conditions on the
numbers $a_n$ has the form of a family of inequalities, each depending on
a finite set of $a_n$, they are preserved by passing to such kind of
limit. Let $\Xi$ be the limit of a sequence of block factor processes, it
seems impossible to prove that it is also a block factor. However, the
corresponding function $h(z)$ will be meromorphic again (!) --- this
follows from the full form of the Carleman inequality, including an upper
estimate of $|h(z)|$ through the coordinates of the poles.
The standard construction of 2-block factor sequence of 0's and 1's is as
follows: $\xi_i=\delta(\eta_i,\eta_{i+1})$, where $\eta$ is an
independent sequence distributed uniformly on $[0,1]$ and $\delta$ is the
characteristic function of a measurable subset
$\Delta\subset[0,1]\times[0,1]$.
Let $H$ be the Hilbert space $H=L_2[0,1]$, $\,v\in H$ be vector
corresponding to the constant function 1 on $[0,1]$, and
$T:H\longrightarrow H$ be the Hilbert--Schmidt operator with the kernel
$\delta$.
Then the probabilities $a_i=P\{\xi_1=\dots=\xi_{i-1}=1\}$ can be expressed
as $a_i=(v,T^{i-1}v)$, so the series is
$h(z)=\sum a_iz^i=1+z(v,(1-zT)^{-1}v)$.
Since $T$ is a compact operator, $(1-zt)^{-1}$ is a meromorphic
operator-valued function on the complex plane.
Let $p_s$ be the sequence of coordinates of its poles, i.~e., $p_s^{-1}$
are nonzero eigenvalues of $T$, repeated counting multiplicities.
At first, the Carleman inequality claims that $\sum |p_s|^{-2}\le \|T\|$.
Therefore, the product $\phi(z)=\prod_s (1-z/p_s)e^{z/p_s}$ is analytic
on the whole plane.
Secondly, it is claimed that
$$
|\phi(z)(1-zT)^{-1}|\le e^{{1\over2}(1+\|T\|^2|z|^2)},
$$
where $\|\cdot\|$ denotes the Hilbert--Schmidt norm and $|\cdot|$ the
uniform norm.
I just rewrote this from Dunford--Schwartz v.~II, XI.6.27, p.~1038.
It seems to be rather clear that the set of all meromorphic functions
satisfying this inequality (with $\|T\|$ fixed) is ``compact'' in an
appropriate sense, that is, any sequence of such functions has a convergent
subsequence.
It follows immediately that the coefficient-wise limit of a sequence of
2-block factor series $h(z)$ is also meromorphic and satisfies the same
inequalities.
Let $\Xi=(\xi_i : i\in Z)$ be a 1-dependent stationary stochastic
sequence of 0's and 1's, and let
$$a_n=P\{\xi_1=...=\xi_{n-1}=1\}, \qquad a_1=a_0=1.$$
Consider the power series $h(z)=\sum a_iz^i$; obviously, it converges
(at least) when $|z|<1$. We have found that it is {\it meromorphic}
for $|z|<2$. Moreover, the series $(1+h(z))^{-1}$ converges in this double
size circle. Indeed, the coefficients of the last series are just the
probabilities like $P\{\sum_{i=1}^n \xi_i\ {\rm is\ even}\}$ with a
multiplier like $2^{-n}$.
Conjecture. For any 1-dependent stationary stochastic sequence $\Xi$,
the series $h(z)$ is meromorphic on the whole complex plane. Moreover,
the sum [of the inverse squares of absolute values of the coordinates
of the poles] $\sum_{z:h(z)=\infty}|z|^{-2}$ (counting multiplicities)
is bounded by a constant.
For the block factor sequences, it is true---as well as I understand---%
and it is written in the book by Danford and Schvarz, the chapter
concerning Hilbert-Shmidt operators (Carleman inequalities).
Furthermore, there is a Tikhonov topology on the set of all $\Xi$:
$\Xi^{(s)}\rightarrow\Xi$ if the corresponding probabilities
$a_n^{(s)}\rightarrow a_n$ for any $n$. Since the conditions on the
numbers $a_n$ has the form of a family of inequalities, each depending on
a finite set of $a_n$, they are preserved by passing to such kind of
limit. Let $\Xi$ be the limit of a sequence of block factor processes, it
seems impossible to prove that it is also a block factor. However, the
corresponding function $h(z)$ will be meromorphic again (!) --- this
follows from the full form of the Carleman inequality, including an upper
estimate of $|h(z)|$ through the coordinates of the poles.
The standard construction of 2-block factor sequence of 0's and 1's is as
follows: $\xi_i=\delta(\eta_i,\eta_{i+1})$, where $\eta$ is an
independent sequence distributed uniformly on $[0,1]$ and $\delta$ is the
characteristic function of a measurable subset
$\Delta\subset[0,1]\times[0,1]$.
Let $H$ be the Hilbert space $H=L_2[0,1]$, $\,v\in H$ be vector
corresponding to the constant function 1 on $[0,1]$, and
$T:H\longrightarrow H$ be the Hilbert--Schmidt operator with the kernel
$\delta$.
Then the probabilities $a_i=P\{\xi_1=\dots=\xi_{i-1}=1\}$ can be expressed
as $a_i=(v,T^{i-1}v)$, so the series is
$h(z)=\sum a_iz^i=1+z(v,(1-zT)^{-1}v)$.
Since $T$ is a compact operator, $(1-zt)^{-1}$ is a meromorphic
operator-valued function on the complex plane.
Let $p_s$ be the sequence of coordinates of its poles, i.~e., $p_s^{-1}$
are nonzero eigenvalues of $T$, repeated counting multiplicities.
At first, the Carleman inequality claims that $\sum |p_s|^{-2}\le \|T\|$.
Therefore, the product $\phi(z)=\prod_s (1-z/p_s)e^{z/p_s}$ is analytic
on the whole plane.
Secondly, it is claimed that
$$
|\phi(z)(1-zT)^{-1}|\le e^{{1\over2}(1+\|T\|^2|z|^2)},
$$
where $\|\cdot\|$ denotes the Hilbert--Schmidt norm and $|\cdot|$ the
uniform norm.
I just rewrote this from Dunford--Schwartz v.~II, XI.6.27, p.~1038.
It seems to be rather clear that the set of all meromorphic functions
satisfying this inequality (with $\|T\|$ fixed) is ``compact'' in an
appropriate sense, that is, any sequence of such functions has a convergent
subsequence.
It follows immediately that the coefficient-wise limit of a sequence of
2-block factor series $h(z)$ is also meromorphic and satisfies the same
inequalities.
