| Popis: |
Let B n ( m ) be a set picked uniformly at random among all m-elements subsets of { 1 , 2 , … , n } . We provide a pathwise construction of the collection ( B n ( m ) ) 1 ⩽ m ⩽ n and prove that the logarithm of the least common multiple of the integers in ( B n ( ⌊ m t ⌋ ) ) t ⩾ 0 , properly centered and normalized, converges to a Brownian motion when both m , n tend to infinity. Our approach consists of two steps. First, we show that the aforementioned result is a consequence of a multidimensional central limit theorem for the logarithm of the least common multiple of m independent random variables having uniform distribution on { 1 , 2 , … , n } . Second, we offer a novel approximation of the least common multiple of a random sample by the product of the elements of the sample with neglected multiplicities in their prime decompositions. |
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