Is there any work on Brownian motion with decaying steps?
For example variations such as:
(1) a random walk in which nth step is $+\frac{1}{\sqrt{n}}$ or $-\frac{1}{\sqrt{n}}$ length of equal probability.
(2) a random walk where nth step has possibilities of $+\frac{1}{\sqrt{n}}, 0, -\frac{1}{\sqrt{n}}$ each of 1/3 probability.
I am interested in finding the effect - if any - such variations have on the growth rate, which for +/-1 is given by:
$$ \limsup_{n \to \infty} \frac{S_n}{\sqrt{2n\log\log n}} = 1 $$
Intutively I feel, the first modification should result in: $$ \limsup_{n \to \infty} \frac{S_n}{\sqrt{2\log\log n}} = 1 $$ and second may result in: $$ \limsup_{n \to \infty} \frac{2S_n}{3\sqrt{2\log\log n}} = 1 $$ but I would like to look at some proofs in this area.