Nate River
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Are $\left[\begin{matrix}x_\ell \\ x_\ell\varphi_k^\ell\end{matrix}\right]$ linearly independent?
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Edit: In response to asker comment, edited significatively the answer as it failed to take into account the case $n<k$ Here is my argument why the probability of linear independence is 1: Case $k\...

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Number of intervals needed to cross, Brownian motion
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For 2. Claim: $\lim_{n\rightarrow\infty}2^{-n/2}\mathbb{E}[K_n]=0$ The idea is this: The union of the events $E_{jn}$ is the event \begin{equation} F_n=\left\{B_t=0 \text{ for some $t\in\left[\...

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