Conditions are given for a randomly indexed sequence of random variables to converge weakly. The key concept employed is the so‐called generalized Anscombe condition. The results give a method of determining sequential stopping rules, which have the required accuracy of estimation of an unknown parameter in the case when the observations are not necessarily independent and identically distributed. Copyright

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Keywords large‐sample theory of sequential estimation, martingales, mixing limit theorems, randomly indexed sequences, stable limit theorems, Weak convergence
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Journal Canadian Journal of Statistics
Csörgo, M, & Rychlik, Z. (1981). Asymptotic properties of randomly indexed sequences of random variables. Canadian Journal of Statistics, 9(1), 101–107. doi:10.2307/3315300