In the 8th chapter of An Introduction to Probability Theory and its Applications. Vol 1 (by William Feller) there are two problems that I'm stuck with, and since the two are interconnected, I decided to ask a question about them at one place. The problems 7 and 8 (both about unending Bernouli sequance) of the aforementioned chapter are as follows:
- Let the $\phi$(t) be a positive monotonically increasing function, and let $n_r$ be the nearest integer to $e^{r/log\,r}$. If $$\sum {\frac{1}{\phi(n_r)}}{e^{-\frac{1}{2}\phi^2(n_r)}}\qquad\qquad(1)$$ converges, then with probability one, the inequality $$S_n>np\,+\sqrt{npq}\,\,\phi(n)\qquad\qquad(2)$$ takes place for finitely many n. Note that without loss of generality we may suppose that $\phi(n)\,<10\sqrt{log\,log\,n}$; the law of the iterated logarithm takes care of the larger $\phi(n)$.
- Prove that the series (1) converges if, and only if,$$\sum\frac{\phi(n)}{n} e^{-\frac{1}{2}\phi^2(n) }\qquad\qquad(3)$$ converges. Hint: Collect the terms for which $n_{r-1}\,<\,\,n\,\,<n_r$ and note that $n_r \,- n_{r-1} \sim n_r(1-1/log\,r) $; furthermore, (3) can converge only if $\phi^2(n)\,\,>\,2\,log\,log\,n$.
[Before going into the challenges that I faced, I should tell you that I am not familiar with measure theory and in the first volume of this book everything is proved and demonstrated without use of it. So I suppose it is possible to solve the problems without measure theory.]
For the problem 7, if we define the sequence of events {$A_r$}s such that the inequality $$S_{n_r}>n_rp\,+\sqrt{n_rpq}\,\,\phi(n_r)$$ holds, and if we assume $\phi(n)$ is adequately small so that normal approximation still applies, say $\phi(n)\,=\,O(\sqrt[7]n)$, we can use the approximation of CDF of normal distribution for large values of x (see here), that is: $$\int_x^{\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\,\simeq\,\frac{1}{x\sqrt{2\pi}}e^{-\frac{x^2}{2}}$$ Now we know the probability assigned to each {$A_r$} for sufficiently large r is the summand of (1) times $1/\sqrt{2\pi}$. As a result we are in a position to apply the first Borel-Cantelli lemma and conclude {$A_r$} s will happen only finitely. But the problem is we only proved the statement for ns of the kind $e^{r/log\,r}$, nothing was said about other ns and the problem is still unsolved. The "Note" which was written at the end of the problem, too, does not help much. It can only help for the set of $\phi(n)$s that satisfy $\lim_{n\to \infty}\frac{\phi(n)}{\sqrt{2\,log\,log\,n}}\,=\,\lambda\,>\,1$. It is ineffective with $\phi(n)$s of the kind: $$\phi(n) =\,(1+\epsilon(n))\, \sqrt{2\,log\,log\,n}$$ in which epsilone tends to zero as n tends to infinity (It does not work on finer forms of the law of the iterated logarithm).
For the problem 8, it seems there is a typo in the hint; the approximation should be: $n_r\,-n_{r-1}\,\sim\,\frac{n_r}{log\,(r-1)}\,\sim\frac{n_r}{log\,r}$. Again, I'm stuck and cannot demonstrate how the two series are related. Even the hint is not helpful to me, I can't relate the summation of terms for ns between $n_{r-1}$ and $n_r$ to the summation (1).
Thank you all.