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 $Z_0, Z_1,...$?
Brownian motion and Fourier series
7
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stochastic-processes
fourier-series
1 Answers
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That's Karhunen-Loève decomposition of Gaussian process.
Check wiki