CLT: having a set of N independent random variables x_i with any probability distribution with mean m_i and standard-deviation s_i thenX_N = Sum[ x_i - m_i ] / sqrt{ Sum[ s_i^2 ] } converges to a Normal random variable as N-> infinityin particular if all m_i=m and all s_i=s, then we have X_N -> N( m, s/sqrt{N})this is useful for cases in finance where hedging is hard or impossible (e.g with stochastic vol)having N trades we can use CLT in the trading book and choose a bid-ask interval based upon our risk-aversion

I was under the impression that CLT issum (x_i - mean) = zeroFurther, in your example, should Sum[ x_i - m_i ] be equal to zero?

hetero you are confusing the central limit thm with the law of large numbers,NB central limit theorem doesnt hold if the random variables have infinite variacneMJ

One clarification. How do you define "independent"?Assume for example you do a monte carlo simulation to price a plaincall option. You have your initial stock price So, which after say N pathsends up being S. You simulate that say M times and you get M payoffs.Are those independent (since they have same So, srike price X, thestock price follows say a lognormal process and the wiener a normal one) ? Could you provide a practical example whereby the CLT is used in finance(if not in the above case) ?Thanks,Apollon.

Well, here goes; the Generalized Central Limit Theorem:X is the limit in distribution of normalized sums of the formS(n) = (X(1) + .... + X(n))/(a(n) - b(n)), where X(1), ... , X(n), whereX are iid, n --> infinity, and X must be a stable distribution.What are a and b????A RV has a stable distribution iff a1*X1 + a2*X2 = a*X + b. where Xs are identical distributions.Examples are:Gaussian (b=0) and exponential.What's key is that the sum over all linear combinations, sum j of a(j))*X(j), is also stable with the same distribution parameters as X(j).Examples of the sums are:sub-Gaussian and Poisson

Indipendence: if a sample of observations is independent, identically distributed (iid) then the pdf can be written as p(x/theta)=product f(x/theta) with x = x1,x2...xn And all xi are independentIf we define X1,X2,…Xn as items of a random sample of size n from any distribution that has mean ì and variance sigma^2 the random variable:Y= (sum Xi )-nì/ sigma*sqrtnHas a limiting Gaussian distribution as n tends to infinite with zero mean and variance=1 The immediate application of CLT in finance are indeed Monte Carlo simulations, if future payoffs of derivatives depend on distribution of returns and prices of underlying , being the number of trials very large (n infinite trials) and simulating a sample average of derivatives payoffs CLT gives a confidence interval to estimate these Payoffs and the std error is proportional to 1/sqrt n .Empirically, another implication of CLT could be distributions of returns calcutated on stock indexes (DJI, S&P).the more you aggregate returns (from daily to yearly), the more their distribution tends to Gaussian.

QuoteEmpirically, another implication of CLT could be distributions of returns calcutated on stock indexes (DJI, S&P).the more you aggregate returns (from daily to yearly), the more their distribution tends to Gaussian. Untrue. The aggregate of sub-Gaussian distributions never tends to be Gaussian.

If I had my copy of Durrett handy I'd look up exactly how strong the independence has to be... I guess I'd say the main implication for financial applications of CLT come from the contrapositive use... you observe a bunch of results that are not anything like normal, and you conclude that there is some correlation buried in the underlying process. And, as we all know, getting these hidden corrs right in the modelling makes the difference between looking like a genius and looking like an idiot.Yeah, probably the stock answer to this question is that unless correlation is present in a model's many microscopic errors, they will tend to cancel eachother out and produce a small, normally distributed noise term whose net variance continues to decline over time. This allows you to keep your models as simple as you can and still get the answer more-or-less correct. You may discover only after you get blown up that there was a hidden tail correlation that caused everything to go so horribly wrong that you never get to play again, which should remind you that CLT is about a limiting distribution and not a stopping time.

What are implications for the central limit theorem in finance ??I'd say:The size of your wallet is inversely proportional to your assumption of iid and useof the central limit theorem.

Too easy answer Newton, and who told you returns for limited intervals returns distributions are sub-gaussian and for the wallet..?Be rigorous do not joke

Mrbadguy,Actually, I wasn't joking at all. Time-series analysis (not that easy) is the key!Good luck,