On the law of the iterated logarithm for subsequences for a stable subordinator (Q1917612)

Let \(\{X(t):0\leq t<\infty\}\) be a stochastic process with stationary independent increments and the characteristic function \(f\) of \(X(1)\) be given by \[ \log f(t)=-|t|^\alpha\{1-i\text{ sign}(t)\tan(\pi\alpha/2)\} \] with \(0<\alpha<1\). The main result of the paper is as follows: a. For \(p>1\) we have \[ \limsup_{k\to\infty} [t^{-1/\alpha}_k X(t_k)]^{1/\log\log t_k}=e^{1/p\alpha}\quad\text{a.s.} \] b. For \(0<p\leq 1\) we have \[ \limsup_{k\to\infty} [t^{-1/\alpha}_k X(t_k)]^{\log\log t_k}=e^{1/\alpha}\quad\text{a.s.}, \] where \(t_k=e^{k^p}\), \(k\geq 1\), \(p>0\) and \(c(\alpha)=\{2B(\alpha)\}^{(1-\alpha)/\alpha}\), and \(B(\alpha)\) is taken from the formula \[ P(X(1)\leq x)\sim(2/\alpha)^{1/2}P(U\geq (2B(\alpha)))^{1/2}x^{-\alpha/(2(1-\alpha))}, \] where the random variable \(U\) has the standard normal distribution.