Moment Convergence in Conditional Limit Theorems
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چکیده
Consider a sum P N 1 Y i of random variables conditioned on a given value of the sum P N 1 X i of some other variables, where X i and Y i are dependent but the pairs (X i ; Y i) form an i.i.d. sequence. We prove, for a triangular array (X ni ; Y ni) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing ; we give also some other applications to occupancy problems, random forests, and branching processes.
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