在這篇論文當中，我們探討對象為一個柯西型隨機漫步。由於其跳躍分佈遵循的柯西型分佈之期望值不存在，這樣的性質導致在分析此分佈時會有額外困難，因此在文獻中較少被探討。我們確立了h個獨立同分佈之一維柯西型隨機漫步的總碰撞次數的極限分佈。更加嚴謹地說，我們證明了以下極限：
\frac{2
i\tanh(
i)}{\log N}\sum_{1\leq i <j\leq h}\mathds{L}^{(i,j)}_N \overset{d}{\Longleftarrow} \Gamma (\frac{h(h-1)}{2},2),
當中L(i,j)_N:= \sum_{n=1}^N\mathds{1}_{\{S^{(i)}_n = S^{(j)}_n\}}為第(i, j)對的隨機漫步之互相碰撞次數，而Γ則為伽傌分佈。這個結果給出了在[ET60]所證明的一個經典定理，並且最近於[CSZ23]和[LZ24]被推廣的一個柯西型隨機漫步版本。

In this thesis, we investigate a random walk that lies within the Cauchy do-main of attraction, an area often overlooked in the literature because of the irregularities in its increment distribution. We identify the limiting distribution of the total pairwise collisions between h i.i.d. one-dimensional Cauchy random walks starting at the origin. Specifically, we establish that
\frac{2
i\tanh(
i)}{\log N}\sum_{1\leq i <j\leq h}\mathds{L}^{(i,j)}_N \overset{d}{\Longleftarrow} \Gamma (\frac{h(h-1)}{2},2),
where L(i,j)_N:= \sum_{n=1}^N\mathds{1}_{\{S^{(i)}_n = S^{(j)}_n\}} is the collision local times of independent copies i and j of the random walk, and \Gamma denotes the Gamma distribution. This provides an analogous result for the Cauchy random walk of a classical theorem that is obtained by [ET60], which is recently generalized in [CSZ23] and [LZ24].

Abstract
Acknowledgement
Section 1 Introduction--------------1
Section 2 Main results--------------6
Section 3 Local limit theorem--------------10
Section 4 Auxiliary results--------------17
Section 5 Chaos expansions of moments--------------24
Section 6 Integral inequalities--------------31
Section 7 Proofs of main results--------------40
Appendix Large Deviations results --------------45

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