论文标题
循环产物和最佳循环出生和死亡链中的最佳陷阱
Cyclic products and optimal traps in cyclic birth and death chains
论文作者
论文摘要
出生死亡链是整数上的一个离散时间马尔可夫链，其过渡概率$ p_ {i，j} $在且仅当$ | i-j | = 1 $时才是非零的。我们认为其出生概率$ p_ {i，i+1} $形成一个定期序列，因此$ p_ {i，i+1} = p_ {i \ mod m} $，对于某些$ m $和$ p_0，\ ldots，p_ {m-1} $。此类链的轨迹$（x_n）_ {n = 0,1，\ ldots} $满足大数字的强大定律和中心限制定理。我们研究重新排序概率$ p_0，\ ldots，p_ {m-1} $对速度$ v = \ lim_ {n \ to \ infty} x_n/n $的效果。 $ v $的标志不受重新排序的影响，但总体上是。我们表明，对于Lebesgue，几乎每一个选择$（p_0，\ ldots，p_ {m-1}）$，恰好可以通过重新排序获得$（m-1）！/2 $不同的速度。我们明确地猜想了将速度降至最低的订购，并证明了所有$ M \ m \ leq 7 $。这种猜想是由我们认为具有独立利益的纯粹组合猜想所暗示的。
A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.
