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Strong tightness as a condition of weak and almost sure convergence

Grzegorz Krupa, Wiesław Zieba (1996)

Commentationes Mathematicae Universitatis Carolinae

A sequence of random elements { X j , j ∈ J } is called strongly tight if for an arbitrary ϵ > 0 there exists a compact set K such that P ⋂ j ∈ J [ X j ∈ K ] > 1 - ϵ . For the Polish space valued sequences of random elements we show that almost sure convergence of { X n } as well as weak convergence of randomly indexed sequence { X τ } assure strong tightness of { X n , n ∈ ℕ } . For L 1 bounded Banach space valued asymptotic martingales strong tightness also turns out to the sufficient condition of convergence. A sequence of r.e. { X n , n ∈ ℕ } is said to converge essentially with...

The "Thirty-seven Percent Rule" and the secretary problem with relative ranks

Béla Bajnok, Svetoslav Semov (2014)

Discussiones Mathematicae Probability and Statistics

We revisit the problem of selecting an item from n choices that appear before us in random sequential order so as to minimize the expected rank of the item selected. In particular, we examine the stopping rule where we reject the first k items and then select the first subsequent item that ranks lower than the l-th lowest-ranked item among the first k. We prove that the optimal rule has k ~ n/e, as in the classical secretary problem where our sole objective is to select the item of lowest rank;...

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