  This paper investigates the symbol error probability~(SEP) of point-to-point
massive multiple-input multiple-output (MIMO) systems using equally likely PAM,
PSK, and square QAM signallings in the presence of transmitter correlation. The
receiver has perfect knowledge of the channel coefficients, while the
transmitter only knows first- and second-order channel statistics. With a
zero-forcing~(ZF) detector implemented at the receiver side, we design and
derive closed-form expressions of the optimal precoders at the transmitter that
minimizes the average SEP over channel statistics for various modulation
schemes. We then unveil some nice structures on the resulting minimum average
SEP expressions, which naturally motivate us to explore the use of two useful
mathematical tools to systematically study their asymptotic behaviors. The
first tool is the Szeg\"o's theorem on large Hermitian Toeplitz matrices and
the second tool is the well-known limit: $\lim_{x\to\infty}(1+1/x)^x=e$. The
application of these two tools enables us to attain very simple expressions of
the SEP limits as the number of the transmitter antennas goes to infinity. A
major advantage of our asymptotic analysis is that the asymptotic SEP converges
to the true SEP when the number of antennas is moderately large. As such, the
obtained expressions can serve as effective SEP approximations for massive MIMO
systems even when the number of antennas is not very large. For the widely used
exponential correlation model, we derive closed-form expressions for the SEP
limits of both optimally precoded and uniformly precoded systems. Extensive
simulations are provided to demonstrate the effectiveness of our asymptotic
analysis and compare the performance limit of optimally precoded and uniformly
precoded systems.
