Brownian law of the iterated logarithm at infinity

For $n$ with $t_n>e$ put $U_n:=\{M_{t_n}>\sqrt{2\beta t_n\log \log t_n}\}$ and $z_n:=\sqrt{2\beta \log \log t_n}$; by [F2] and [F3], $P(U_n)=2\bar{\mathrm{\Phi }}(z_n)\le 2\phi (z_n)\mathrm{/}{z}_n\le C_{\alpha ,\beta }{n}^{-\beta }(\log n{)}^{-1\mathrm{/}2}$ for a constant $C_{\alpha ,\beta }$, because $\phi (z_n)=(2\pi {)}^{-1\mathrm{/}2}(\log t_n{)}^{-\beta }=(2\pi {)}^{-1\mathrm{/}2}(n\log \alpha {)}^{-\beta }$; since $\beta >1$ the probabilities are summable. For example, grouping $n$ in $[2^j,2^{j+1})$ bounds the upper series by a constant times ${\sum }_j{2}^{j(1-\beta )}$, which is geometric. Set the finitely many early events with $t_n\le e$ to the empty event.

For the lower bound fix $\alpha >1$, put $\beta :=\alpha \mathrm{/}(\alpha -1)>1$ and $D_n:=B_{t_{n+1}}-B_{t_n}$, so that by [F1] the $D_n$ are independent with law $N(0,t_{n+1}-t_n)=N(0,t_{n+1}\mathrm{/}\beta )$; let $E_n:=\{D_n>\gamma \sqrt{2t_{n+1}\log \log t_{n+1}}\}$ with $\gamma :=1\mathrm{/}\sqrt{\beta }$, and note ${\gamma }^2\beta =1$.

By [F4] and [step 1.1] there is a probability-one event on which $U_n$ fails for all sufficiently large $n$; on that event, for every $t\ge t_N$ with $N$ large, choosing $n$ with $t_n\le t<t_{n+1}$ gives $M_t\le M_{t_{n+1}}\le \sqrt{2\beta t_{n+1}\log \log t_{n+1}}$ and hence $\frac{M_t}{\sqrt{2t\log \log t}}\le \sqrt{\beta \alpha \frac{\log \log (\alpha t)}{\log \log t}}\to \sqrt{\alpha \beta }$; therefore ${\mathrm{lim\hspace{0.17em}sup}}_{t\to \mathrm{\infty }}\frac{B_t}{\sqrt{2t\log \log t}}\le \sqrt{\alpha \beta }$ almost surely.

For large $n$ the probability of $E_n$ satisfies $P(E_n)=\bar{\mathrm{\Phi }}(\gamma \sqrt{2\beta \log \log t_{n+1}})\ge \frac{1}{2}{z}_n^{-1}\phi (z_n)=\frac{1}{2\sqrt{2\pi }}\frac{1}{\sqrt{2\log \log t_{n+1}}\log t_{n+1}}$, with $z_n=\sqrt{2\log \log t_{n+1}}$, by the lower Mills bound of [F3]; since $\log t_{n+1}=(n+1)\log \alpha$, for sufficiently large $n$ this is at least $c_{\alpha }\mathrm{/}((n+1)\sqrt{\log (n+1)})$ with $c_{\alpha }>0$. In each block $2^j\le n+1<2^{j+1}$, the sum of these lower bounds is at least a positive constant times $1\mathrm{/}\sqrt{j+1}$; hence the series diverges (grouping that latter series into square blocks already gives a fixed positive contribution per block). Define the finitely many early events with $t_{n+1}\le e$ to be empty.

Intersecting the events of [step 2.1] over the countably many pairs of rationals $\alpha ,\beta \in (1,2)$ and using [F6] to choose, for every $\epsilon >0$, rationals with $\sqrt{\alpha \beta }\le 1+\epsilon$, we obtain ${\mathrm{lim\hspace{0.17em}sup}}_{t\to \mathrm{\infty }}\frac{B_t}{\sqrt{2t\log \log t}}\le 1$ almost surely.

By [F5] the process $-B$ is again a standard Brownian motion, so [step 3.1] applies to it and gives ${\mathrm{lim\hspace{0.17em}inf}}_{t\to \mathrm{\infty }}\frac{B_t}{\sqrt{2t\log \log t}}\ge -1$ almost surely.

By [F4] and [step 2.2] the events $E_n$ occur infinitely often almost surely; on the event of [step 4.1] intersected with this one, for infinitely many $n$ one has $B_{t_{n+1}}=B_{t_n}+D_n\ge -(1+\epsilon )\sqrt{2t_n\log \log t_n}+\gamma \sqrt{2t_{n+1}\log \log t_{n+1}}=\sqrt{2t_{n+1}\log \log t_{n+1}}(\gamma -(1+\epsilon ){\alpha }^{-1\mathrm{/}2}(1+o(1)))$.

Since $\gamma =\sqrt{1-1\mathrm{/}\alpha }\to 1$ and ${\alpha }^{-1\mathrm{/}2}\to 0$ as $\alpha \to \mathrm{\infty }$, for every $\delta >0$ there are rational $\alpha >1$ and $\epsilon >0$ with $\gamma -(1+\epsilon ){\alpha }^{-1\mathrm{/}2}\ge 1-\delta$; hence [step 5.1] gives ${\mathrm{lim\hspace{0.17em}sup}}_{t\to \mathrm{\infty }}\frac{B_t}{\sqrt{2t\log \log t}}\ge 1-\delta$ almost surely for every rational $\delta >0$, and intersecting over the countably many $\delta$ yields $\mathrm{lim\hspace{0.17em}sup}\ge 1$ almost surely.

Combining [step 3.1] and [step 6.1] gives ${\mathrm{lim\hspace{0.17em}sup}}_{t\to \mathrm{\infty }}\frac{B_t}{\sqrt{2t\log \log t}}=1$ almost surely; applying this conclusion to the standard Brownian motion $-B$ of [F5], whose limit superior is the negative of the limit inferior of $B$, gives ${\mathrm{lim\hspace{0.17em}inf}}_{t\to \mathrm{\infty }}\frac{B_t}{\sqrt{2t\log \log t}}=-1$ almost surely.

The boundary and degeneracy cases are covered: the normalizer is positive precisely for $t>e$, and the statement is asymptotic as $t\to \mathrm{\infty }$; the geometric sequences are indexed from $n=0$, with only finitely many terms below $e$; the parameters over which probability-one events are intersected may be restricted to rational $\alpha ,\beta >1$ and rational $\delta >0$, a countable family. The auxiliary value $\gamma =\sqrt{1-1\mathrm{/}\alpha }$ need not be rational and creates no additional event: once $\alpha$ is fixed it is a deterministic threshold in the same block events $E_n$. The upper and lower bounds are established using the first Borel-Cantelli lemma for the upper bound and the second for the lower bound; AC enters only through [F7].