Central limit theorem for the Walsh system
It is well-known that a lacunary orthogonal system of functions bears similar properties to the independent random variables. This paper investigates a problem which remained open in a paper of the reviewer, Stud. Sci. Math. Hung. 10(1975), 141-146 (1977; Zbl 0365.60031), concerning the validity of the central limit theorem in case of strongly lacunary Walsh systems. The author proves that under the lacunarity condition \[ n_{k+1}/n_ k\geq 1+c/\sqrt{k}\quad (c>0,\quad \alpha =1/2) \] there exists a sequence \(\{n_ k\}\) satisfying the above condition such that for the subsystem \(\{W_{n_ k}(x)\}\) of the Walsh functions the central limit theorem is not valid.
scientific article; zbMATH DE number 3942609