why acf for stationary process is positive semi definite I am studying the time series, and the lecture note just state that
For the autocovariance function of a stationary time series one of the property is positive semi definite .
I want to see the proof, but I cant google it
 A: kjetil b halvorsen's comment is witty and accurately pointed out the necessity. In any case, this is a routine procedure covered by any good time series textbook. Nevertheless, for the sake of completeness, I am jotting down (paraphrasing) the proof, which basically is the characterization of acv functions.

Kolmogorov Consistency Theorem: Let $\{\mathbf F_{\mathbf t}, ~\mathbf t := \left(t_1,t_2,\ldots,t_n)^\mathsf T\in \mathbb R^n, ~t_1<t_2<\cdots<t_n,~~n\in \mathbb N\right\}$ be the set of finite dimensional distribution functions. $\bf F_t$s can be the distribution functions of a stochastic process if and only if $$\lim_{x_i \to \infty} \mathbf F_{\mathbf t}(\mathbf x) = \mathbf F_{\mathbf t(i)} (\mathbf x(i)),$$ $\mathbf t(i)$ is $\mathbf t$ sans the $i$ th component.
It would take more of a space to derive this and it would, frankly, derail from the matter in hand. Nonetheless, here is discussed elegantly the proofs both for countable as well as uncountable products.

Theorem: A function $\gamma: \mathbb Z \to \mathbb R$ is acv function of a stationary time series if and only if it is even and non-negative definite.
I would adumbrate the proof here.
Let $\mathbf a\in \mathbb R^n,~ \mathbf t \in \mathbb Z^n$ and let $\mathbf Z_{\mathbf t} := (X_{t_1}- \mathbb EX_{t_1},\ldots, X_{t_n}- \mathbb EX_{t_n})^\mathsf T. $ Since $\mathbb V(\mathbf a^\mathsf T \mathbf Z_{\mathbf t}) \geqslant 0$ and $\mathbb V(\mathbf a^\mathsf T \mathbf Z_{\mathbf t}) = \mathbf a^\mathsf T \mathbb E \mathbf Z_{\mathbf t}\mathbf Z_{\mathbf t}^\mathsf T\mathbf a, $ one can conclude $[\gamma(t_i-t_j) ]_{i, ~j~=~1}^n$ is non-negative definite.
For the converse, the approach would be to show the existence of a stationary process that is associated with an even non-negative definite function $\gamma(\cdot),$ the latter being the acv of the former. Let $\mathbf K:= [\gamma(t_i-t_j) ]_{i, ~j~=~1}^n.$ For each $n\in \mathbb N, ~\mathbf t\in \mathbb R^n, :t_1<t_2<\cdots<t_n,$ let $\mathbf F_{\mathbf t}$ be the distribution function  in $\mathbb R^n$ with characteristic function
$$ \varphi_{\mathbf t}(\mathbf u) := \exp\left(-\mathbf u^\mathsf T\mathbf K\mathbf u/2\right),~\mathbf u\in \mathbb R^n.$$ By assumption, $\mathbf K$ is non-negative and this means $\varphi_{\mathbf t}(\cdot) $ is the characteristic function of an $n$-variate normal distribution with mean $\bf 0$ and covariance matrix $\bf K. $ Now, the critical thing is to note that
$$\varphi_{\mathbf t(i)}(\mathbf u(i)) = \lim_{\mathbf u(i) \to 0}\varphi_{\mathbf t}(\mathbf u) , \forall\mathbf t. $$ This implies $\mathbf F_{\mathbf t}$ is Kolmogorov consistent and hence there exists a time series $\langle X_t\rangle$ with the distribution function $\mathbf F_{\mathbf t}.$

Reference:
Time Series: Theory and Methods, Peter J. Brockwell, Richard A. Davis, Springer Science & Business Media, 1991.
