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By a result of Heinrich, Novak, Wasilkowski and Wo\'zniakowski the inverse ofthe star discrepancy $n(d,\varepsilon)$ satisfies $n(d,\varepsilon)\leqc_{\abs}d\varepsilon^{-2}$. Equivalently for any $N$ and $d$ there exists a setof $N$ points in $[0,1)^d$ with star discrepacny bounded by$\sqrt{c_{\abs}\cdot d/N}$. They actually proved that a set of independentuniformly distributed random points satisfies this upper bound with positiveprobability. Although Aistleitner and Hofer later refined this result byproving a precise value of $c_{\abs}$ depending on the probability with whichthe inequality holds, so far there is no general construction for such a set ofpoints known. In this paper we consider the sequence $(x_n)_{n\geq 1}=(\langle2^{n-1}x_1\rangle)_{n\geq 1}$ for a uniformly distributed point $x_1\in[0,1)^d$ and prove that the star discrepancy is bounded by$C\sqrt{d\log_2d/N}$. The precise value of $C$ depends on the probability withwhich this upper bound holds.

Probabilistic Star Discrepancy Bounds for Lacunary Point Sets