In the seventies, Charles Stein revolutionized the way of proving the Central Limit Theorem by introducing a method that utilizes a characterization equation for Gaussian distribution. In the last 50 years, much research has been done to adapt and strengthen this method to a variety of different settings and other limiting distributions. However, it has not been yet extended to study conditional convergences. In this article, we develop a novel approach using Stein's method for exchangeable pairs to find a rate of convergence in Conditional Central Limit Theorem of the form (Xn | Yn = k), where (Xn, Yn) are asymptotically jointly Gaussian, and extend this result to a multivariate version. We apply our general result to several concrete examples, including pattern count in a random binary sequence and subgraph count in Erdös-Rényi random graph.
Consider a collaborative dynamic of k independent random walks on a finite connected graph G. We are interested in the size of the set of vertices visited by at least one walker and study how the number of walkers relates to the efficiency of covering the graph. To this end, we show that the expected size of the union of ranges of k independent random walks with lifespans t1, t2, . . . , t k , respectively, is greater than or equal to that of a single random walk with the lifespan equal to t1 + t2 + • • • + t k . We analyze other related graph exploration schemes and end with many open questions.
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