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Lets say I have an fair coin.

Lets say we have a random variable X, and X=1 if the coin is heads, and X=-1 if the coin is tails.

Now lets define a random process Y(t)=X. This means that there is one coin toss, and it defines the value of Y(t).

Is Y(t) SSS? It seems that it should be, because its PDF isn't dependent in time.

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    What is SSS? Please define it. – 2017-02-07
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    Yes, this process is strictly stationary. It is not _ergodic_ in the sense that the time average $\lim_{T\to \infty}\frac 1T \int_{0^T} y(t) dt$ of a sample path $y(t)$ (there are only two of these!) does not converge to the ensemble average $E[Y(t)] = E[X]$. – 2017-02-07
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    SSS is Strict Sense Stationary – 2017-02-07

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If you mean strongly (or strictly) stationary then the answer is yes. For arbitrary $k$ and any $t \in \mathbb{R}_+^k$ we have that $$[Y(t_1),\ldots,Y(t_k)] = [X,\ldots,X].$$ Similarly, for $\tau > 0$ we have that $$[Y(t_1+\tau),\ldots,Y(t_k+\tau)] = [X,\ldots,X].$$ Combining these two equations yields the desired result.