I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 A. Lyapunov central limit theorem.
 B. Lindeberg-Feller central limit theorem.
 5 Random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 11 Functional Analysis.
 12 Fourier analysis.
 13 Sobolev spaces.
 14 Elliptic PDE.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Central limit theorem (CLT) II.

he CLT was introduced on elementary level in the section ( Central limit theorem ). The calculation of that section has restrictive assumptions and the result lacks generality. Here we consider the issue with greater precision.

Let be a family of r.v. We assume that for each fixed the family is independent. This chapter studies limiting distributions of as .

We denote and the ch.f. and cumulative distribution functions of respectively.

We think of sums as quantities of similar magnitude even though the number of terms in these sums increases. We formalize such view in the following definition.

Definition

(Holospoudic) The family is called "holospoudic" if the following condition holds

Proposition

(Holospoudic criteria). The family is holospoudic iff

Proof

First, we assume that is true and proceed to prove the statement .

We estimate The last term tends to zero under the assumption and .

Next, we assume that and prove .

We invoke the proposition ( P.m. vs ch.f. estimate ): where , : or The desired statement follows from the above by the proposition ( Dominated convergence theorem ).

 A. Lyapunov central limit theorem.
 B. Lindeberg-Feller central limit theorem.
 Notation. Index. Contents.