B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

448 FUNDAMENTALS OF PROBABILITY THEORY
Now is a good time to further relax the conditions in the central lirnittheorem for the sample
mean. This highly important generalization is proved by the famous Russian mathematician
A. Lyapunov in 1901.
Central Limit Theorem (for the sum of independent random variables):
Let random variables X J, .•. , Xn be independent but not necessarily identically distributed.
Each of the random variable Xi has mean /.,li and nonzero variance a? < oo. Furthermore,
suppose that each third-order central moment
and suppose
i = I, ... , n
Then random variable
converges to a standard Gaussian density as n ➔ oo, that is,
lim P [y(n) > x] = Q(x) (8.99)
n➔oo
The central limit theorem provides a plausible explanation for the well-known fact that
many random variables in practical experiments are approximately Gaussian. For example,
communication channel noise is the sum effect of many different random disturbance sources
(e.g., sparks, lightning, static electricity). Based on the central limit theorem, noise as the sum
of all these random disturbances should be approximately Gaussian.
REFERENCES
1. J. Singh, Great Ideas of Modern Mathematics, Dover, Boston, 1959.
2. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions, National Bureau of
Standards, Washington, DC, 1964, sec. 26.
3. The Chemical Rubber Co., CRC Standard Mathematical Tables, 26th ed., 1980.
4. J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering, Wiley, New York,
1965, p. 83.
5. N. S. Jayant and P. Noll, Digital Coding of Wa veforms: Principles and Applications to Speech and
Video, Prentice-Hall, 1984.
6. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed., McGraw-Hill, New
York, 1995.
7. M. H. DeGroot, Probabilities and Statistics, 2nd ed., Addison Wesley, Reading, MA, 1987.