Confusion about speech being piecewise stationary From wikipedia:
"Modern general-purpose speech recognition systems are based on Hidden Markov Models. These are statistical models that output a sequence of symbols or quantities. HMMs are used in speech recognition because a speech signal can be viewed as a piecewise stationary signal or a short-time stationary signal. In a short time-scale (e.g., 10 milliseconds), speech can be approximated as a stationary process. Speech can be thought of as a Markov model for many stochastic purposes."

I get that if you break a speech signal into small enough parts, then a every part will make a very specific, basic sound, which I presume, has the same (or very similar) type of spectral envelope, as many other parts around it/close to it.. but I don't quite understand what the stationary process is, especially not in this context.

To be exact, I don't understand what exactly the stochastic process/it's random variables are, or even the motivation for describing a piece of signal as a stochastic process. 
I mean the actual input is not a stochastic process, it's a concrete input, it's an observation - is the quote saying that basically any small enough piece of the input can be considered as a certain sample of some stationary process? 

Do we call it stationary, because while we know that the small part has some sort of quality of being very repetitive, we don't know exactly where we've seperated it/where it starts? Similar to something like a stationary process of sin(t+Y), Y being a random variable.

I'm pretty confused about this whole thing- if I see a specific sample of white noise, is it correct to call it 'stationary'? I mean, the original distribution that generated this sample, that is a stationary process, but does that mean you call the sample/signal itself 'stationary'?

How exactly does the fact that sound is piecewise stationary imply it a good idea to use HMM? I simply get stuck at that - maybe a slow, thorough explanation of this implication (and what exactly it means that it's piecewise stationary) would be the most helpful to me.

Sorry if the questions are not understandable or not concrete enough. I'm simply very confused and don't know enough about this.
In this context, stationary usually means that the spectral density is (theoretically) the same throughout the observation window.
So, piecewise stationary means that if you divide the signal into short enough time frames, the signal in each frame can be described, accurately enough, as sum of sinusoids in presence of white noise (which is stationary as well).