论文标题
非负2 BY-2矩阵的渐近产品，并应用于随机步行的渐近零漂移
Asymptotics of product of nonnegative 2-by-2 matrices with applications to random walks with asymptotically zero drifts
论文作者
论文摘要
令$ a_ka_ {k-1} \ cdots a_1 $为某些非负2 by-2矩阵的产物。通常，它的要素很难评估。在某些情况下，我们表明$ \ forall i，j \ in \ {1,2 \}，$ $（a_ka_ {a_ka_ {k-1} \ cdots a_1）_ {i，j} \ sim c \ sim c \ sim c \ varrho（a_k） $ k \ rightArrow \ infty，$ \ varrho（a_n）$是矩阵$ a_n $和$ c \ in（0，\ infty）$的光谱半径，因此可以估算$ a_ka_ {ka_ {ka_ {ka_ \ cdots a_1 $的元素。作为应用，请考虑（2,1）和（1,2）随机步行的某些偏移的最大值，其渐近零漂移。 我们获得了一些微妙的限制理论，这些理论与简单的随机步道完全不同。限制尾部和关键尾部序列的理论在我们的研究中起着重要作用。
Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim c\varrho(A_k)\varrho(A_{k-1})\cdots \varrho(A_1)$ as $k\rightarrow\infty,$ where $\varrho(A_n)$ is the spectral radius of the matrix $A_n$ and $c\in(0,\infty)$ is some constant, so that the elements of $A_kA_{k-1}\cdots A_1$ can be estimated. As applications, consider the maxima of certain excursions of (2,1) and (1,2) random walks with asymptotically zero drifts. We get some delicate limit theories which are quite different from the ones of simple random walks. Limit theories of both the tail and critical tail sequences of continued fractions play important roles in our studies.
