Probability Theory and Random numbers : Gap Theoretical Approach

设^2_0为函数f的一类∫^1_0 f(x)dx=0,∫^1_0 |f(x)|^2 dx <∞。对于给定递增整数序列{n_k}和给定函数类X∧L^2_0，我们setΨ[X；{n_k}] (t) = < lim一口> ___ < K→∞> <一口> ___ < f∈X >(Σ^ K_ < K = 1 > f (n_kt)) /(√< K日志日志K >)我们调查这个函数的值分布。我们证明了Ψ[{f}；{n_k}]【小于或等于】< | | f | | _A f =Σ{| ^ ^ ^ (v) |乙醯。在高桥间隙条件下，n_<k+1>/n_k【大于等于】> 1 + c/k^β(c >0,β<1/2)。我们也证明了这个结果的统一版本：Ψ[X；{n_k}][小于或等于]sup||f||_A a.e.对于某些条件下的X。

Let ^2_0 be the class of function f with ∫^1_0 f(x)dx=0,∫^1_0 |f(x)|^2 dx < ∞.For given sequence {n_k} of increasing integers and given class X⊂ L^2_0 of functions, we setΨ[X;{n_k}](t)=<lim sup>___<K→∞><sup>___<f∈X> (Σ^K_<k=1>f(n_kt))/(√<K log log K>)We investigated the value distribution of this function.We proved Ψ[{f};{n_k}]【less than or equal】< ||f||_A =Σ{|f^^^(v)|a.e. under Takahashi's gap condtion n_<k+1>/n_k 【greater than or equal】> 1 + c/k^β(c >0,β<1/2). We also proved the uniform version of this result :Ψ[X;{n_k}]【less than or equal】sup||f||_A a.e.for X of some conditions.