论文标题
rencontre问题
The rencontre problem
论文作者
论文摘要
令$ \ left \ {x^{1} _k \ right \} _ {k = 1}^{\ infty}，\ left \ left \ {x^{2} _k \ \ right \} _ {k = 1} \左\ {x^{x^{d} _K \ \ right \} _ {k = 1}^{\ infty} $ be $ d $独立序列的bernoulli随机变量，带有成功范围$ p_1，p_1，p_2，p_2，p_2，p_2，\ cdots，p_d $ $ d $ d $ $ d $ $ 0 <是$ dee <is a p $ n is aj y s aj y是$ quq 2 $，<是$ 0 < $ j = 1,2，\ cdots，d。$ let \ begin {equation*} s^{j}（n）= \ sum_ {i = 1}^{n}^{n} x^{j} _ {j} _ {i} = x^= x^{ x^{j} _ {n}，\ Quad n = 1,2，\ cdots。 \ end {equation*}我们在时间$ n $上声明一个“ rencontre”，或者等效地，说$ n $是“ rencontre time”，如果\ begin {equation*} s^{1} {1} {1}（n）= s^{2} = {2}（n）= \ cdots = \ cdots = s^d} d} d}（n）（n）（n）。 \ end {equation*}我们激励和研究第一个（提供有限的）rencontre时间的分布。
Let $\left\{X^{1}_k\right\}_{k=1}^{\infty}, \left\{X^{2}_k\right\}_{k=1}^{\infty}, \cdots, \left\{X^{d}_k\right\}_{k=1}^{\infty}$ be $d$ independent sequences of Bernoulli random variables with success-parameters $p_1, p_2, \cdots, p_d$ respectively, where $d \geq 2$ is a positive integer, and $ 0<p_j<1$ for all $j=1,2,\cdots,d.$ Let \begin{equation*} S^{j}(n) = \sum_{i=1}^{n} X^{j}_{i} = X^{j}_{1} + X^{j}_{2} + \cdots + X^{j}_{n}, \quad n =1,2 , \cdots. \end{equation*} We declare a "rencontre" at time $n$, or, equivalently, say that $n$ is a "rencontre-time," if \begin{equation*} S^{1}(n) = S^{2}(n) = \cdots = S^{d}(n). \end{equation*} We motivate and study the distribution of the first (provided it is finite) rencontre time.
