an application of taylor' theorem in the neighborhood of yields psz (tn ps ( ps ( ( / since ps ( dtd eitz = ez and ps ( ez - we have# in psn (tpsz (tn ( / - - / from example we recognize - / as the characteristic function of the standard normal distribution thusfrom theorem we conclude that - ( figure shows the central limit theorem in action the left part shows the pdfs of for the case where the {xi have [ distribution the right part shows the same for the exp( distribution in both caseswe clearly see convergence to bell-shaped curvecharacteristic of the normal distribution figure illustration of the central limit theorem for (leftthe uniform distribution and (rightthe exponential distribution the multivariate version of the central limit theorem is the basis for many asymptotic (in the size of the training setresults in machine learning and data science theorem multivariate central limit theorem let xn be iid random vectors with expectation vector and finite covariance matrix define xn :( xn )/ thenm-estimator (xn - ( sas one application is as follows suppose that parameter of interestthis the unique solution of the system of equations ps( th where ps is vector-valued (or multivaluedfunction and the distribution of does not depend on th an -estimator of th