Probability theory debunked.

The Wiener process aka Brownian motion is defined as a stochastic process on the positive reals with Gaussian independent increments and $W_0 = 0$, This means $W_t -W_s$ has to be independent of $W_s - W_0 = W_s$ for any [math[ s \leq t [/math].

Now by the Karhunen-Loeve expansion

https://en.wikipedia.org/wiki/Karhunen-loeve_expansion#The_Wiener_process

there exists the representation $W_t = \sqrt{2} \sum_{k=1}^{\infty} Z_k \frac{sin((k-\frac{1}{2})\pi t)}{ (k-\frac{1}{2})\pi}$
where $Z_k$ are indpendent Gaussian random variables with mean 0 and variance 1.

Does this already look suspicious? Indeed, how can a process with time parameter be "random" if it merely consists of deterministic trajectories of random variables whose outcome is already fixed at t = 0 ? In particular the increments cannot be random and most importantly not independent.

The independence of increments of the Wiener process therefore yields
$0 = Cov(W_s, W_t - W_s) = 2 \sum_{k=1}^{\infty} \frac{sin((k-\frac{1}{2})\pi s)}{ (k-\frac{1}{2})\pi} (\frac{sin((k-\frac{1}{2})\pi t)}{ (k-\frac{1}{2})\pi} - \frac{sin((k-\frac{1}{2})\pi s)}{ (k-\frac{1}{2})\pi})$

If this is to hold for all [math[ s \leq t [/math] the partial derivative w.r.t. t has to vanish when we keep s < t fixed:
$0 = \partial_s Cov(W_s, W_t - W_s) = 2 \sum_{k=1}^{\infty} \frac{sin((k-\frac{1}{2})\pi s)cos((k-\frac{1}{2})\pi t)}{ (k-\frac{1}{2})\pi}$

Taking the limit $t \to s$ and using trigonometric identies:
$0 = \sum_{k=1}^{\infty} \frac{sin( 2 (k-\frac{1}{2})\pi s)}{ (k-\frac{1}{2})\pi}$

which is clearly bullshit for $s \neq 0$.

Since the Karhunen-Loeve theorem was derived by valid methods of mathematical proof, the only conclusion is that the theory of stochastic processes is complete and utter nonsense and should be disregarded.