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Let $(B_t)_{t \in [0, \infty)}$ be a Brownian motion. Can you prove me why it can be written as $$B_t= Z_0 \cdot t + \sum_{k=1}^{\infty} Z_k \frac{\sqrt{2} \cdot \sin(k \pi t)}{k \pi}$$ for some independent standard normal random variables $Z_0, Z_1,...$?

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That's Karhunen-Loève decomposition of Gaussian process.

Check wiki

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