Abstract
Let \(\{X_n,n\ge 1\}\) be a sequence of independent and identically distributed random variables with a common distribution function F is in the domain of partial attraction of a semistable law with index \(\alpha \), \(0< \alpha < 2\). We introduce new concept delayed random sums and study a non-trivial limit behavior of properly normalized subsequences of delayed random sums and extended to some boundary crossing problem.
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