What i meant by Taylor series of a polynomial is that that a polynomial can serve as a Taylor series of a function. Thus, my question is spread not only over polynomials but also over functions which are defined by bounded number of digits because of the natural constraint of boundedness. The latter says that all processes in Nature happen at specific for each process, yet bounded rates and so that not to exceed specific, also bounded thresholds. Thus, in my opinion, happens coupling of truncation and round-off error in natural processes.

It is not that I or anybody else wants to truncate a series. According to my view truncation is a natural process. And since the corresponding polynomial is defined up to bounded number of digits also naturally, due to boundedness, this is what I call natural round-off. 

Only in this sense a polynomial can serve as a "Taylor series".  I mean that a bounded function can be approximated by appropriate  polynomial to its value at a neighborhood.

What is important is that, due to the existence of thresholds so that all variations happen within them, there exists average of a time series according to Lindeberg theorem (see Feller)  and I guess that the latter must be result of decoupling of truncation and round-off.

I offer to call the above truncation and round-off "natural" dropping "errors" since they appear as a result of natural processes.

More generally the problem is how their interplay affects the average.

I am afraid that you do not quite understand the general physical setting I am talking about. The major idea is that all natural processes must be bounded in order that a system to stay stable. This immediately yields finite number of digits for defining any function that represents a process. As a consequence all trajectories intersects which is in radical contradiction with the well known rules for  differential equations and trajectories in the state space in statistical mechanics.

In this line of suggestions appears the problem with defining "Taylor" series of the functions that define a process. In order to meet the requirement about finite number of digits it is necessary to suggest  that  a "Taylor" series comprises only bounded number of members. Thus, "round-off", i.e. the number of significant digits, turns entangled with the \\'truncation". So far, this is a generic property of any function whose general Taylor series converges.

My question is is there any other constraint necessary  for providing existence of a mean and variance for each bounded sequence as proven by Feller. 

In order to make my idea clearer to you I offer you to read my book "Boundedness and Self-Organized Semantics: Theory and Applications" published by IGI-Global,  Hershey, PA, USA.  Maybe then you would understand my problem.

The round-off error is smaller than the truncation error with half orders of magnitude.

I will try to explain myself in a conversation. Yet, I think that a systematic reading of the book is of much help.

My experience teaches me that a damping  ball does not bounce infinitely many times before resting. This is because each swing takes finite time no matter how small it is. In my model there are no infinitesimals. Alongside, I do not use convergence of series or integrals because the latter involves  very small numbers which require an infinite number of significant digits; and this contradicts the major idea about boundedness of accumulation and exchange of matter/information/energy.

As about the Big Bang: my theory is not a  theory of everything it is about the behavior of complex systems. One of the enigmas in their behavior is their remarkable stability which survives in an ever-changing environment.

About the coefficients in polynomials: in my theory they appear as Taylor coefficients and are defined with bounded precision.

Classical mechanics is a model, not reality. Mine is also model, but I hope it is closer to reality than the classical mechanics.

And yes, I do propose some changes in the traditional calculus but still they are minimal and not fully developed. That is what my question is about. My goal is to develop appropriate mathematical apparatus for the theory. I lot of analytic work has been done in the book but there is still a lot to come.