TY - JOUR

T1 - The game of normal numbers

AU - Lehrer, Ehud

PY - 2004/5

Y1 - 2004/5

N2 - We introduce a two-player game where at each period one player, say, Player 2, chooses a distribution and the other player, Player 1, chooses a realization. Player 1 wins the game if the sequence of realized outcomes is normal with respect to the sequence of distributions. We present a pure winning strategy of Player 1 and thereby provide a universal algorithm that generates a normal sequence for any discrete stochastic process. It turns out that to select the nth digit, the algorithm conducts O(n2) calculations. The proof uses approachability in infinite-dimensional spaces (Lehrer 2002).

AB - We introduce a two-player game where at each period one player, say, Player 2, chooses a distribution and the other player, Player 1, chooses a realization. Player 1 wins the game if the sequence of realized outcomes is normal with respect to the sequence of distributions. We present a pure winning strategy of Player 1 and thereby provide a universal algorithm that generates a normal sequence for any discrete stochastic process. It turns out that to select the nth digit, the algorithm conducts O(n2) calculations. The proof uses approachability in infinite-dimensional spaces (Lehrer 2002).

KW - Normal number

KW - Two-player game

UR - http://www.scopus.com/inward/record.url?scp=4043088370&partnerID=8YFLogxK

U2 - 10.1287/moor.1030.0087

DO - 10.1287/moor.1030.0087

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AN - SCOPUS:4043088370

VL - 29

SP - 259

EP - 265

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

SN - 0364-765X

IS - 2

ER -