Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities.

Given a realisation $\omega$ of the random variables, we say that $X_i (\omega)$ is a running maximum if $X_i (\omega) \geq X_j(\omega)$ for all $j < i$.

Question: Let $M_n$ denote the number of running maximums up to time $n$. Is it true that

$$M_n \sim \log n$$

almost surely as $n \to \infty$?

That is, do there exist deterministic constants $c, C > 0$ such that for almost every $\omega$ and for all large enough $N$, possibly depending on $\omega$, we have

$$c \log n \leq M_n (\omega) \leq C \log n$$

for all $n \geq N(\omega)$?

Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities.

Given a realisation $\omega$ of the random variables, we say that $X_i (\omega)$ is a running maximum if $X_i (\omega) \geq X_j(\omega)$ for all $j < i$.

Question: Let $M_n$ denote the number of running maximums up to time $n$. Is it true that

$$M_n \asymp\log n$$

almost surely as $n \to \infty$?

That is, do there exist deterministic constants $c, C > 0$ such that for almost every $\omega$ and for all large enough $N$, possibly depending on $\omega$, we have

$$c \log n \leq M_n (\omega) \leq C \log n$$

for all $n \geq N(\omega)$?

Let $\{X_i\}_{i \geq 1}$ be a sequence of iid uniform random variables on $[0, 1]$. Given a realisation $\omega$ of the random variables, we say that $X_i (\omega)$ is a running maximum if $X_i (\omega) \geq X_j(\omega)$ for all $j < i$.

Question: Let $M_n$ denote the number of running maximums up to time $n$. Is it true that

$$M_n \sim \log n$$

almost surely as $n \to \infty$?

That is, do there exist deterministic constants $c, C > 0$ such that for almost every $\omega$ and for all large enough $N$, possibly depending on $\omega$, we have

$$c \log n \leq M_n (\omega) \leq C \log n$$

for all $n \geq N(\omega)$?

Let $\{X_i\}_{i \geq 1}$ be a sequence of iid non atomic random variables, that is, their CDF has no jump discontinuities. 

Given a realisation $\omega$ of the random variables, we say that $X_i (\omega)$ is a running maximum if $X_i (\omega) \geq X_j(\omega)$ for all $j < i$.

Question: Let $M_n$ denote the number of running maximums up to time $n$. Is it true that

$$M_n \sim \log n$$

almost surely as $n \to \infty$?

That is, do there exist deterministic constants $c, C > 0$ such that for almost every $\omega$ and for all large enough $N$, possibly depending on $\omega$, we have

$$c \log n \leq M_n (\omega) \leq C \log n$$

for all $n \geq N(\omega)$?

Let $\{X_i\}_{i \geq 1}$ be a sequence of iid uniform random variables on $[0, 1]$. Given a realisation $\omega$ of the random variables, we say that $X_i (\omega)$ is a running maximum if $X_i (\omega) \geq X_j(\omega)$ for all $j < i$.

Question: Let $M_n$ denote the number of running maximums up to time $n$. Is it true that

$$M_n \sim \log n$$

almost surely as $n \to \infty$?

That is, do there exist deterministic constants $c, C > 0$ such that for all large enough $N$, possibly depending on $\omega$, we have

$$c \log n \leq M_n (\omega) \leq C \log n$$

for all $n \geq N(\omega)$?

Let $\{X_i\}_{i \geq 1}$ be a sequence of iid uniform random variables on $[0, 1]$. Given a realisation $\omega$ of the random variables, we say that $X_i (\omega)$ is a running maximum if $X_i (\omega) \geq X_j(\omega)$ for all $j < i$.

Question: Let $M_n$ denote the number of running maximums up to time $n$. Is it true that

$$M_n \sim \log n$$

almost surely as $n \to \infty$?

That is, do there exist deterministic constants $c, C > 0$ such that for almost every $\omega$ and for all large enough $N$, possibly depending on $\omega$, we have

$$c \log n \leq M_n (\omega) \leq C \log n$$

for all $n \geq N(\omega)$?