上临界Galton-Watson树轮廓函数的 Scaling 极限研究

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本文主要探讨了Galton-Watson树的轮廓函数在极限情况下的行为，特别是在上临界状态下的特征。Galton-Watson树是一种随机树模型，它的分支结构基于每个个体的子代数量，具有重要的统计和概率论意义。上临界和下临界是两种不同的生长模式，上临界树指的是其平均子代数大于1，而下临界树则相反。作者何辉和栾娜娜在文章中回顾了Abraham和Delmas近期的工作，他们利用鞅变换（martingale transformation）这一工具，成功地建立了上临界Lévy树和下临界Lévy树之间的关联。Lévy树是一种具有特定自相似性的随机树，它们在自然界和金融等领域有广泛的应用。通过这种转换，他们能够构造出在固定高度上的上临界Lévy树的分布，这对于理解这些复杂系统的性质至关重要。在离散的Galton-Watson树的背景下，文章指出这种类似的连接同样适用。作者利用对下临界树轮廓函数收敛性研究的现有成果，进一步证明了截断的上临界Galton-Watson树的轮廓函数在弱收敛意义上趋近于Abraham和Delmas构建的分布。这里的轮廓函数是指沿着树的边缘路径定义的函数，它反映了树的形状和结构。本文的关键概念包括概率论与数理统计、Galton-Watson树、分支过程、Lévy树以及Scaling极限。Scaling极限是数学物理学中的一个重要概念，它关注的是随着尺度变化，系统的行为如何趋于稳定或特定形式。在这个上下文中，它指的是一种极限过程，使得当树的高度趋向于无穷大时，轮廓函数的性质能够被精确地描述和预测。本文的核心内容是深入探讨了如何通过鞅变换来理解和刻画上临界Galton-Watson树在固定高度下的统计特性，并且通过比较和连接上、下临界树的轮廓函数，揭示了这种随机树在不同生长状态下行为的连续性和差异性。这对于理论研究和实际应用都具有重要意义。

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needed to explore the tree is ζ(t) := 2(#t − 1). The contour function of t is the function

(C(t, s), 0 ≤ s ≤ ζ(t)) whose value at time s ∈ [0, ζ(t)] is the distance (on the tree) between

the position of the particle at time s and the root. We set C(t, s) = 0 for s ∈ [ζ(t), 2#t].

Given a probability distribution p = {p

: n = 0, 1, . . .} with p

< 1, following [17] and [4],

call a T

∞

-valued random variable G

a Galton-Watson tree with oﬀspring distribution p if

i) the number of children of ∅ has distribution p:

P(k

∅

= n) = p

, ∀ n ≥ 0;

ii) for each h = 1, 2, . . . , conditionally given r

= t with |t| ≤ h, for ν ∈ t with |ν| = h,

are i.i.d. random variables distributed according to p.

The second property is called branching property. From the deﬁnition we have for a ∈ N

∗

and

any t ∈ T such that |t| ≤ a,

P(r

= t) =



ν∈t:|ν|<a

. (1)

Deﬁne m(p) =



k≥0

. We say G

is sub-critical (resp. critical, super-critical) if m(p) <

1(resp. m(p) = 1, m(p) > 1).

Given a tree t ∈ T

∞

, then r

t is a ﬁnite tree whose contour function is denoted by

(t, s) : s ≥ 0}. For k ≥ 0, we denote by Y

(t) the number of individuals in generation k:

(t) = #{ν ∈ t : |ν| = k}, k ≥ 0.

Given a probability measure p = {p

: k ≥ 0} with



k≥0

> 1. Let G

be a super-

critical Galton-Watson tree with oﬀspring distribution p. Then P(#G

< ∞) = f(p), where

f(p) is the minimal solution of the following equation of s:

(s) :=



k≥0

= s, 0 ≤ s ≤ 1.

Let q = {q

: k ≥ 0} be another probability distribution such that

= f(p)

k−1

, for k ≥ 1, and q

= 1 −



k≥1

Then



k≥0

< 1. Let G

be a subcritical GW tree with oﬀspring distribution q. Note

that (Y

), k ≥ 0) is a Galton-Watson process starting from a single ancestor with oﬀspring

distribution q. We ﬁrst present a simple lemma.

Lemma 1. Let F be any nonnegative measurable function on T. Then for t ∈ T,

P [G

= t] = f (p)P [G

= t] (2)

and for any a ∈ N,

E [F(r

)] = E



f(p)

1−Y

)

F (r

)



. (3)

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上临界Galton-Watson树轮廓函数的 Scaling 极限研究

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