Concept of LPC

Given a digital system, can the value of any sample be predicted by taking a linear combination of the previous $N$ samples?

Stated mathematically, can a set of coefficients, $a_k$, be determined such that

$$y(n) = a_1y(n-1) + a_2y(n-2) + ... + a_Ny(n-N).$$

That is, can a signal be represented as coefficients of an all-pole (only feedback terms) IIR filter.

If yes, then the coefficients and the first $N$ samples would completely determine the remainder of the signal, because the rest of the samples can be calculated by the above equation.