Suppose that $\{X_n\}_n$ is an i.i.d. sequence of random variables. Put $S_n:= \sum_{j=1}^n X_j$ and assume that
$$\limsup_{n \to \infty} \frac{|S_n|}{\sqrt{n\log \log n}}< \infty\quad a.s.$$
Put $X_j':= X_{2j}-X_{2j-1}, j \geq1$ and $S_n' := \sum_{j=1}^n X_j'$. Can we deduce that
$$\limsup_{n \to \infty} \frac{|S_n'|}{\sqrt{n\log \log n}}< \infty\quad a.s.$$
Attempt: This is used in the proof of the converse of the law of the iterated logarithm, so I can't definitely use that the second moments exist as this is the end goal.
I tried to arrange $|S_n'| = |\sum_{j=1}^n X_{2j}- \sum_{j=1}^n X_{2j-1}|$ so I can tell estimate it with |S_n| or something like that but I'm clearly overseeing something.
Let $U_i=X_{2i}$ and $V_i=X_{2i-1}$. Then $(U_i)$ is equivalent to $(X_i)$ in the sense the two sequeces have the same finite dimensional distributions. Similarly, $(V_i)$ is equivalent to $(X_i)$. Hence $\lim \sup \frac {|T_n|}{\sqrt {n\log \log n}} <\infty$ a.s. and $\lim \sup \frac {|Z_n|}{\sqrt {n\log \log n}}<\infty$ a.s where $T_n$ an d $Z_n$ are the partial sums of $(U_n)$ and $(V_n)$. This implies your result.