Probability Theory and Random numbers : Gap Theoretical Approach

设^2_0是函数f的类，满足&lt;$^1_0 f（x）dx=0，&lt;$^1_0| f（x）|对于给定的递增整数序列{n_k}和给定的函数类X &lt;$L^2_0，我们设[X;{n_k}]（t）=<lim sup>__&lt;K→∞&gt;<sup>__&lt;f∈X&gt;（&lt;$^K_&lt;k=1&gt;f（n_kt））/（t<K log log K>）.我们研究了这个函数的值分布，证明了[{f};{n_k}][小于或等于]&lt;||F||_A ={|f^（v）|A.E.在Takahashi能隙条件n_&lt;k+1&gt;/n_k [大于或等于]&gt; 1 + c/k^β（c &gt;0，β&lt;1/2）下.我们还证明了这个结果的统一形式：&lt;$[X;{n_k}][小于或等于]sup|| F||_A a.e.对于某些条件的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.