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Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with $$ P(X_i=1)=P(X_i=-1)=1/2, $$ and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As is well known, the sum $S_n$ is (null) recurrent and satisfies the law of the iterated logarithm $$ P\left(\limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1\right)=1, $$

Assume now that we restrict ourselves to the subset of realisations of $S_n$ that are all surely recurrent and where each realisation satisfies $\frac{1}{n} S_n \to 0$. The law of the iterated logarithm obviously holds also in this case, but can it be strengthened further? In other words:

Q: If $S_n$ is surely recurrent and satisfies $\frac{1}{n} S_n \to 0$, does this imply $$ \limsup_{n\to \infty} \frac{|S_n|}{\sqrt{2n \log \log n}}\leq 1? $$

Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with $$ P(X_i=1)=P(X_i=-1)=1/2, $$ and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As is well known, the sum $S_n$ is (null) recurrent and satisfies the law of the iterated logarithm $$ P\left(\limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1\right)=1, $$

Assume now that we restrict ourselves to the subset of realisations of $S_n$ where each realisation satisfies

1. $\frac{1}{n} S_n \to 0$.

2. For any integer $m$, there are infinitely many values of $n$ such that $S_n=m$ or $S_n=-m$.

Does the law of the iterated logarithm take a stronger form in this case, in the sense:

Q: If $S_n$ is satisfies the criteria above, does this imply $$ \limsup_{n\to \infty} \frac{|S_n|}{\sqrt{2n \log \log n}}\leq 1? $$

Or are there even in this case subsets of realisations for which this is not true?

Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with $$ P(X_i=1)=P(X_i=-1)=1/2, $$ and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As is well known, the sum $S_n$ is (null) recurrent and satisfies the law of the iterated logarithm $$ P\left(\limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1\right)=1, $$

Assume now that we restrict ourselves to the subset of realisations of $S_n$ that are all surely recurrent. The law of the iterated logarithm obviously holds also in this case, but can it be strengthened further? In other words:

Q: If $S_n$ is surely recurrent, does this imply $$ \limsup_{n\to \infty} \frac{|S_n|}{\sqrt{2n \log \log n}}\leq 1? $$

Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with $$ P(X_i=1)=P(X_i=-1)=1/2, $$ and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As is well known, the sum $S_n$ is (null) recurrent and satisfies the law of the iterated logarithm $$ P\left(\limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1\right)=1, $$

Assume now that we restrict ourselves to the subset of realisations of $S_n$ that are all surely recurrent and where each realisation satisfies $\frac{1}{n} S_n \to 0$. The law of the iterated logarithm obviously holds also in this case, but can it be strengthened further? In other words:

Q: If $S_n$ is surely recurrent and satisfies $\frac{1}{n} S_n \to 0$, does this imply $$ \limsup_{n\to \infty} \frac{|S_n|}{\sqrt{2n \log \log n}}\leq 1? $$

Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with $$ P(X_i=1)=P(X_i=-1)=1/2, $$ and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As is well known, the sum $S_n$ is (null) recurrent and satisfies the law of the iterated logarithm $$ P\left(\limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1\right)=1, $$

Assume now that we restrict ourselves to the subset of realisations of $S_n$ that are all surely recurrent. The law of the iterated logarithm obviously holds also in this case, but can it be strengthened further? In other words:

Q: If $S_n$ is surely recurrent, does this imply $$ \limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1? $$

Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X_1, X_2, \dots$ be i.i.d. random variables with $$ P(X_i=1)=P(X_i=-1)=1/2, $$ and define $S_n=X_1+\dots+X_n$ with $S_0=0$. As is well known, the sum $S_n$ is (null) recurrent and satisfies the law of the iterated logarithm $$ P\left(\limsup_{n\to \infty} \frac{S_n}{\sqrt{2n \log \log n}}=1\right)=1, $$

Assume now that we restrict ourselves to the subset of realisations of $S_n$ that are all surely recurrent. The law of the iterated logarithm obviously holds also in this case, but can it be strengthened further? In other words:

Q: If $S_n$ is surely recurrent, does this imply $$ \limsup_{n\to \infty} \frac{|S_n|}{\sqrt{2n \log \log n}}\leq 1? $$