我們研究具有 L 種型態的超臨界多型態分支過程 {Z_n}_{n≥0}。首先，在適當的條件下，探討從第 n 代隨機選取一個個體時，其祖先型態的漸近比例。接著，我們考慮此過程在實數線 R 上所對應的分支隨機漫步。令 Z_{n,i}(x) 表示第 n 代中，位置小於或等於 x 的型態 i 個體數。我們證明存在一發散的數列 {β_n}_{n=0}^∞，使得 Z_{n,i}(β_nx)/|Zn| 以 L^2 的形式收斂。最後，我們證明從第 n代中隨機選取一個個體，其位置在機率分布上收斂至標準常態分布。

We study a supercritical multi-type branching process {Z_n}_{n≥0} with L types. First, we investigate, under suitable conditions, the asymptotic proportion of ancestral types for an individual randomly selected from generation n. Next, we consider the associated branching random walk on the real line R. Let Z_{n,i}(x) denote the number of type-i individuals in generation n whose positions are less than or equal to x. We show that there is a sequence {β_n}_{n=0}^∞ with β_n → ∞ such that the ratio Z_{n,i}(β_nx)/|Zn| converges in L^2. Finally, we establish that the position of a uniformly chosen individual from generation n converges in distribution to the standard normal law.

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