论文标题
在具有给定度序列的多元随机森林中，分支林的总人口和枚举多人森林的枚举
On Multitype Random Forests with a Given Degree Sequence, the Total Population of Branching Forests and Enumerations of Multitype Forests
论文作者
论文摘要
学位序列$（n_ {i，j}（k），1 \ leq i，j \ leq d，k \ geq 0）$ type $ i $的个人数量，具有$ k $ j $ j $ j $ j $。我们用给定度序列（MFGD）统一地构造一个均匀地采样的多人森林。为此，我们使用（Chaumont and Liu，2016）对投票定理的扩展，并将Vervaat变换（Vervaat，1979）推广到多维离散可交换增量过程中。我们证明，MFGD是Multype Galton-Watson（MGW）森林的扩展，因为将前者的定律混合在一起，因此获得了MGW森林，其类型为固定尺寸（CMGW）。我们还通过MGW森林中的类型获得了总人口的法律，从而概括了Otter-Dwass公式（Otter 1949，Dwass 1969）。我们将其应用于获得平面，标记和二元多型森林的枚举。我们提供了一种算法来模拟某些CMGW森林，从而概括了（Devroye，2012年）的Unitype案例。
The degree sequence $(N_{i,j}(k),1\leq i,j\leq d,k\geq 0)$ of a multitype forest with $d$ types, is the number of individuals type $i$, having $k$ children type $j$. We construct a multitype forest sampled uniformly from all multitype forest with a given degree sequence (MFGDS). For this, we use an extension of the Ballot Theorem by (Chaumont and Liu, 2016), and generalize the Vervaat transform (Vervaat, 1979) to multidimensional discrete exchangeable increment processes. We prove that MFGDS are extensions of multitype Galton-Watson (MGW) forests, since mixing the laws of the former, one obtains MGW forests with fixed sizes by type (CMGW). We also obtain the law of the total population by types in a MGW forest, generalizing Otter-Dwass formula (Otter 1949, Dwass 1969). We apply this to obtain enumerations of plane, labeled and binary multitype forests having fixed roots and individuals by types. We give an algorithm to simulate certain CMGW forests, generalizing the unitype case of (Devroye, 2012).
