User - MathOverflow

binomial transform, Hurwitz zeta function

2013-02-24T16:22:58Z

For $j,n\in\mathbb Z_+$, let $$ L_{j,n}^{(t)}= \sum_{m=0}^{n} \Bigl(-\frac 12\Bigr)^{n-m}{n\choose m}{m+j+1\choose m+1} \left( \frac {1}{t+\frac 12}\right)^{m+j+2} $$ and $$ L_{j,n} =\sum_{t=1}^\infty L_{j,n}^{(t)} $$

We would like to show that for $j,n\ge 0$ $$ (*)\quad \quad L_{2j,2n}>0. $$ Hypothesis (*) is supported by the numerics we did. In fact, it seems that $L_{j,n}>0$ is positive for all $j>0$ and all $n$. However, we only need the result for even indexes.

What do we know? Namely, the following easy facts:

$L_{j,0}>0$ for all $j$.
$L_{0,2n}>0$ for all $n$.
$L_{j,n}^{(1)}>0$ for all $n$ and $j>0$.

These easy facts follow from the observation that

$$ L_{j,n}^{(t)}=\frac {(-1)^j}{j!}\frac {d^j}{dz^j} \left( \frac {1}{z^2} \left[ \frac {1}{z}-\frac 12\right]^n \right)\left|_{z=t+\frac 12}\right. $$

In general, we need to estimate the sign of $$ \sum_{t=1}^\infty \Phi_n^{(j)}(t+\frac 12),\quad\text{where } \Phi_n(z)=\frac {1}{z^2} \Bigl[ \frac {1}{z}-\frac 12\Bigr]^n. $$ Heuristically, $$ (-1)^j j! L_{j,n}=\sum_{t=1}^\infty \Phi_n^{(j)}(t+\frac 12) \approx -\Phi_n^{(j-1)}(\frac 32) $$ and the sign of $\Phi_n^{(j)}(z)$ is $(-1)^j$ for every $z<2$, which supports the conjecture as well.

Eigenvectors of contraction times projection

2013-02-12T10:52:09Z

Suppose $A$ is a real $n\times n$ matrix with real eigenvalues: $$ 1=\lambda_1>|\lambda_2|\ge \ldots\ge |\lambda_n|>0. $$ Suppose $B$ is an involution, for simplicity let us assume that $B$ is diagonal, has $k$ ones on the diagonal, and $n-k$ minus ones. Suppose I also know that $AB$ has an eigenvector $ABx=x$. Thus I know that $$ ABx=x,\quad Ay=y $$ for some $y$.

Can I conclude that $A(I+B)/2$ has eigenvalue $1$ as well, ie. $$ A\cdot \frac {I+B}{2} z=z $$ for some $z$?

Estimate entropy of a binary process in terms of decay of correlations

2012-12-25T17:11:41Z

Suppose $( X_{n} )$ is an ergodic binary process with $$ \mathbb P(X_{n}=1)= \mathbb P(X_{n}=0)=\frac 12. $$

Naturally the entropy (rate) $h(X)$ of $X=(X_{n})$ satisfies $$ h(X)=\lim_{n\to\infty} \frac 1n H(X_1,\ldots, X_n)\le H(\frac 12)=\log 2. $$ The entropy will "drop" due to dependencies in $(X_n)$.

Suppose that $$ \sigma^2=\sum_{n=2}^{\infty} \sup_{a,b\in{0,1} }\Bigl| \mathbb P(X_{1}=a,X_{n}=b)-\mathbb P(X_{1}=a)\mathbb P(X_{n}=b)\Bigr| $$ is very small: $\sigma^2\ll 1$. Small $\sigma^2$ suggests that $(X_{n})$ are nearly independent. Can one derive a lower bound for $h(X)$ of the form $$ h(X)\ge \log 2- f(\sigma^2), $$ where $f$ is a continuous function with $f(0)=0$? My feeling is that small $\sigma^2$ should imply that the $\bar{d}$-metric between ${X_{n}}$ and the $\text{Ber}(1/2)$-process should be small, hence implying that entropies should be close as well.

In the oposite direction, suppose $\sigma^2$ is large. Then the entropy entropy drop $\log 2- h(X)$ should be substantial as well. For example, for every $\epsilon>0$ one can find $D>0$ such that for every binary process with $\sigma^2>D$ $$ h(X)\le \log 2 -\epsilon. $$

Any ideas, references?

Answer by waler for Minimum 1st-neghbors distance between N random points on a ring

2012-12-03T12:30:23Z

PYKE, R. (1965). Spacings. J. R. Stat. Soc. Ser. B Stat. Methodol. 27 395–449.

Answer by waler for functions whose average along orbits is zero or a constant

2012-04-22T09:23:50Z

Second part: if your system is nontrivial, and function $f$ is "smooth", than condition of the form $$ \lim_{n\to\infty} \frac 1n\sum_{i=0}^{n-1} f(T^i x) =c $$ for all $\ x$, should imply that $$ f(x)= g(x)-g(Tx)+c $$ for some function $g$. In this case, one says that $f$ is cohomologous to a constant.

Answer by waler for Approximating Moment of Sum of RVs

2012-08-22T16:46:14Z

Take a loot at R. Ibragimov and Sh. Sharakhmetov, The Exact Constant in the Rosenthal Inequality for Random Variables with Mean Zero, Theory Probab. Appl., 46(1), 127–132. (6 pages) Read More: http://epubs.siam.org/doi/abs/10.1137/S0040585X97978762

Abstract Let $\xi_1, \ldots, \xi_n$ be independent random variables with ${\bf E}\xi_i=0,$ ${\bf E}|\xi_i|^t<\infty$, $t>2$, $i=1,\ldots, n,$ and let $S_n=\sum_{i=1}^n \xi_i.$ In the present paper we prove that the exact constant ${\overline C}(2m)$ in the Rosenthal inequality $$ {\bf E}|S_n|^t\le C(t) \max \Bigg(\sum_{i=1}^n{\bf E}|\xi_i|^t,\ \Bigg(\sum_{i=1}^n {\bf E}\xi_i^2\Bigg)^{t/2}\Bigg) $$ for $t=2m,$ $m\in {\bf N},$ is given by $$ \overline C(2m)=(2m)! \sum_{j=1}^{2m} \sum_{r=1}^j \sum \prod_{k=1}^r \frac {(m_k!)^{-j_k}} {j_k!}, $$ where the inner sum is taken over all natural $m_1 > m_2 > \cdots > m_r > 1$ and $j_1, \ldots, j_r$ satisfying the conditions $m_1j_1+\cdots+m_rj_r=2m$ and $j_1+\cdots+j_r=j$. Moreover $$ \overline C(2m)={\bf E}(\theta-1)^{2m}, $$ where $\theta $ is a Poisson random variable with parameter 1.

In your case, one gets $$ {\bf E}|S_n|^{2m} \le \overline C(2m) \max( n, (var S_n)^{m})= \overline C(2m) \max( n, n^{m}c^m). $$ where $c=var(\xi_i)$. Thus for large $m$, $$ \Bigl( {\bf E}|S_n|^{2m}\Bigr)^{\frac 1{2m}} \le (\overline C(2m))^{1/(2m)} \sqrt{n} c. $$ Finally, they cite earlier papers where it was shown that $$ \overline C(t)= O( t/\log t). $$ Ans since is $(t/\log t)^{1/t}<\sqrt{t}$ for $t>2$ we are done.

determinant of diagonal - fixed

2012-04-21T10:23:45Z

I have to study/evaluate many determinants of the form $$ f_M(J)=\det(J-M), $$ where $M$ is fixed, and $J$ is a diagonal matrix (with 0/1 on the diagonal, if it helps.) In my problem $M$ is fixed, and $J$ varies. Any suggestions?

greatest common divisor of two special polynomials

2012-02-19T12:51:34Z

The gcd of $x^n-1$ and $x^m-1$ is $x^{gcd(n,m)}-1$. Is it known what the greatest common divisor of $(x^{n_1}-1)(x^{n_2}-1)$ and $(x^{m_1}-1)(x^{m_2}-1)$ is?

Comment by

2013-02-12T13:01:39Z

Yes, a typo, thank you.

Comment by

2012-12-26T19:48:47Z

Thanks, the process I want to understand has many other useful properties I can use, I was looking for minimal conditions.

Comment by

2012-08-22T18:40:20Z

@Mark Meckes: 1968 edition, chapter 4. dependent variables, section 20: mixing processes, paragraph on moment inequalities.

Comment by

2012-08-22T18:36:21Z

@Bill Johnson: absolutely, severe overkill. result must follow also from some easier inequalities as well.

Comment by

2012-08-22T16:08:23Z

Look at Lemma 4, page 172, in Billingsley's book Convergence of probability measures. Lemma is for p=4, but it works for all even p. If I am not mistaken, this lemma gives the bound you are looking for.

Comment by

2012-04-23T13:31:50Z

the most famous example of a result of this nature is the so-called Livshic lemma: suppose $X$ is a mixing subshift, $T:X\to X$ is a left shift, and $f$ is a Holder-continuous function such that $$ \sum_{i=0}^{p-1} f(T^ix) =0 $$ for every periodic $x$: $x=T^px$. Then $f=g-g\circ T$. There are many generalizations of this result.

Comment by

2012-04-22T09:14:43Z

Thank you very much, this looks quite interesting.