= Solution
Let $S_N=\sum_{n=0}^N\sum_{k=0}^{2^n-1}a_{n,k}H_{n,k}$. The assumed <orthonormality> of the <Haar wavelets> gives
$$
\|S_N\|_{\mathcal H}^2=\sum_{n=0}^N\sum_{k=0}^{2^n-1}a_{n,k}^2.
$$
Enumerate the coefficients level by level as $A_1,A_2,\ldots$. They are <independent and identically distributed> <standard normal random variables>, so the squares have <mean> one. The <strong law of large numbers> implies
$$
\frac1m\sum_{j=1}^mA_j^2\longrightarrow1\quad\text{almost surely}.
$$
Consequently,
$$
\boxed{\|S_N\|_{\mathcal H}^2\longrightarrow\infty\quad\text{almost surely}.}
$$
A convergent sequence in a <Hilbert space> must be bounded in <norm>. Hence this random <Haar wavelet> series diverges in $\mathcal H$ <almost surely>.
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