I have the following problem.
Let $X_1,X_2,...$ be independent random variables with the same continuous distribution. Let $E_n = \{X_n>X_m, \forall m<n\}$ be the event of a new record at time $n$ and $N_n$ the amount of records at time $n$.
What I already did is to show that $E_1, E_2,...$ are independent with $P(E_n) = \dfrac{1}{n}$. So far so good.
But the main goal is to proof $\dfrac{N_n}{log(n)} \to 1$ a.s. The idea is to use Kroneckers Lemma for which I have to show that $\sum_{k=1}^{\infty} \dfrac{I_{E_k} - 1/k}{log(k)}$ converges. I don't know how to approach the latter.
Any help is appreciated.
The idea is to show that there is a martingale and then show by $\mathcal{L}^2$-restriction that it converges almost surely.
First define $X_k=\frac{I_{E_k}-\frac{1}{k}}{log (k)}$. Because $P(E_n)=\frac{1}{n}$ as stated in your question we have $\mathbb{E}[X_k]=0$ $\forall$k. Then you can show with the usual steps that $M_n=\sum_{k=1}^{n}X_k$ is a martingale.
To show $\mathcal{L}^2$-restriction, note that $\mathbb{E}[(M_{k+1}-M_k)^2]=\mathbb{E}[(X_{k+1})^2]\leq\frac{1}{k+1(log(k+1))^2}$
This means you can say $\sum_{k=1}^n\mathbb{E}[(M_{k+1}-M_k)^2]<\infty$ so that $sup_{n\geq 0} \mathbb{E}[M_n^2]<\infty,$ which tells you that $(M_n)$ is a $\mathcal{L}^2$-restricted martingale which converges almost surely according to Doob's martingale theorem.