In this paper,we prove a general law of the iterated logarithm (LIL) for independent non-identically distributed B-valued random variables.As an interesting application,we obtain the law of the iterated logarithm for the empirical covariance of Hilbertian autoregressive processes.
Let an, n≥ 1 be a sequence of independent standard normal random variables. Consider the randomtrigonometric polynomial Tn（θ）=∑^n_i=1 aj cos（j θ）, 0≤θ≤π and let Nn be the number of real roots of Tn（θ）in （0, 2π）. In this paper it is proved that limn→∞ Var（Nn）/n=co,where 0 〈 co〈 ∞.
