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during latest decades , we have seen large numbers of publications on the use of pareto - lev and other heavy - tailed distributions in finance ( see , for example , @xcite , @xcite and references there ) . one of the key authors that initiated this , was mandelbrot .
he mentioned that gaussian distribution can not provide a solid explanation for observed large amount of ` outliers ' - in other words , the number of observations for which absolute value of deviation from empirical mean is bigger than @xmath0 , where @xmath1 is empirical variance .
aparently , mandelbrot considered stable distributions as a unique alternative to gaussian family basing on the fact of classical limit theorem for independent identically distributed ( i.i.d . ) random variables , or , equivalently , on stability property . in the remainder of this paper
, we show that stable distributions can not explain such a large number of outliers either .
the connection to classical limit theorem for a non - random number of random summands seems to be questionable .
a more natural perspective comes from the use of limit theorem for a random number of random variables .
in this section we look at the probability of observing outliers in the case of distributions belonging to a domain of attraction of strictly stable distribution .
specifically , suppose that @xmath2 is a sequence of i.i.d .
random variables .
denote by @xmath3 their empirical mean and empirical variance correspondingly .
let @xmath4 be a fixed number .
we are interested in the following probability @xmath5 our aim here is to prove the following theorem .
[ th1 ] suppose that @xmath2 is a sequence of i.i.d .
random variables belonging to a domain of attraction of strictly stable random variable with index of stability @xmath6 .
then @xmath7    since @xmath8 belong to the domain of attraction of strictly stable random variable with index @xmath9 , it is also true that @xmath10 belong to the domain of attraction of one - sided stable distribution with index @xmath11 .
\1 ) consider at first the case @xmath12 . in this case ,
@xmath13{}{\longrightarrow } a={{\rm i}\kern-0.18em{\rm e}}x_1 $ ] and @xmath14{}{\longrightarrow } \infty$ ] .
we have @xmath15 @xmath16{}{\longrightarrow } 0.\ ] ]    \2 ) suppose now that @xmath17 . in this case
, we have @xmath18 as @xmath19 . here
@xmath20 is @xmath21-stable random variable , and the sign @xmath22 is used for asymptotic equivalence .
similarly , @xmath23 where @xmath24 has one - sided positive stable distribution with index @xmath11 .
we have @xmath25 @xmath26{}{\longrightarrow } 0.\ ] ]    \3 ) in the case @xmath27 we deal with cauchy distribution .
the proof for this case is very similar to that in the case 2 ) .
we omit the details .
let us note that for the case of distributions having finite second moment and non - compact support , the probability ( [ eq1 ] ) has a positive limit as @xmath19 .
indeed , if the second moment is finite then both @xmath28 and @xmath29 have finite limits @xmath30 and @xmath31 .
the probability ( [ eq1 ] ) converges to @xmath32 as @xmath19 because of non - compactness of support of the distribution .
therefore , it is clear that the explanation of presence a large number of outliers can not be provided by the heaviness of the tails .
although the results of section [ s2 ] show that asymptotically the probability of presence of a large number of outliers is negligible , we may still have doubts for not too large samples .
however , the samples in practice are not too small .
so , basing on actual observed data , we can consider the sample size @xmath33 of order @xmath34 as typical one .
we simulated @xmath35 samples of size @xmath33 ( @xmath33 is growing from 1,000 to 25,000 with step 2,000 ) and calculated the estimate of probability @xmath36 given by ( [ eq1 ] ) for @xmath37 .
the behavior of this probability as a function of @xmath33 is reflected on figure [ fig1 ] .
blue line corresponds to symmetric stable distribution with @xmath38 ; red line - to standard gaussian distribution .
we see that the probability @xmath36 for @xmath38-stable symmetric distribution becomes smaller that for gaussian case starting at about @xmath39 .
therefore , we can not expect many outliers for such stable distribution for typical sample size .     for @xmath37 .
blue line corresponds to symmetric stable distribution with @xmath38 ; red line - to standard gaussian distribution ]    \2 .
again , we simulated @xmath35 samples of size @xmath33 ( @xmath33 is again growing from 1,000 to 25,000 with step 2,000 ) and calculated the estimate of probability @xmath36 given by ( [ eq1 ] ) for @xmath40 .
the behavior of this probability as a function of @xmath33 is plotted on figure [ fig2 ] .
blue line corresponds to symmetric stable distribution with @xmath41 ; red line - to standard gaussian distribution .
we see that the probability @xmath36 for @xmath41-stable symmetric distribution becomes smaller than that for gaussian case starting from about @xmath42 .
the decrease of @xmath36 for the case @xmath37 is much slower .
for example , @xmath43 for @xmath41-stable symmetric distribution versus @xmath44 for gaussian distribution .
in this situation , we can not say that stable distribution provides less outliers that gaussian law .
but will the corresponding number of outliers be sufficient for the predicted figures to be in agreement with observed data ?
unfortunately , there are only a few papers giving observed number of outliers , that is an estimate of probability @xmath36 for @xmath37 . some data of this kind may be found in @xcite , table 3 .
corresponding estimates for probabilities @xmath45 given there vary from 0.009 to 0.013 . for symmetric stable distribution with @xmath46 , this probability is @xmath43 .
we see that it is too small to explain the number of outliers in table 3 from @xcite .
let us note that similar simulations for different values of parameters and sample size are given in @xcite .
we shall not discuss them here .     for @xmath40 .
blue line corresponds to symmetric stable distribution with @xmath41 ; red line - to standard gaussian distribution ]
lately , _ tempered stable distributions _ have been growing increasingly popular .
the idea of such distribution lies in altering the tails of stable laws by exponential tails starting from a certain point of distribution s support . of course
, the probability @xmath36 in this case will not converge to zero anymore .
however , one may expect that this probability will be small if the point of the tails alteration is far from the origin , since tempered distribution will be close to stable in this case .
simulations support this opinion .
let us consider only the case of symmetric tempered stable distributions having characteristic functions @xmath47 for @xmath48 , @xmath49 , @xmath50 .    on figure [ fig2a ] , plot of limit probability @xmath51
is given for the distribution ( [ eqa ] ) as a function of parameters @xmath21 and @xmath52 for @xmath53 .
this plot shows that the probabilities are too small to explain the presence of large number of outliers .     as a function of parameters @xmath54 and @xmath55 . ]
here we discuss a way of constructing from a distribution another one having a higher probability to observe outliers .
we call this procedure `` put tail down '' .
let @xmath56 be a probability distribution function of random variable @xmath57 having finite second moment @xmath58 and such that @xmath59 for all @xmath60 . take a parameter @xmath61 and
fix it . define a new function @xmath62 where @xmath63 for @xmath64 , and @xmath65 for @xmath66 .
it is clear that @xmath67 is probability distribution function for any @xmath61 .
of course , @xmath68 also has finite second moment @xmath69 , and @xmath70 .
however , @xmath71 .
let @xmath72 be a random variable with probability distribution function @xmath68 .
then @xmath73 denoting @xmath74 rewrite previous equality in the form @xmath75 for @xmath72 to have more outliers than @xmath57 it is sufficient that @xmath76 there are many cases in which inequality ( [ eq4 ] ) is true for sufficiently large values of @xmath77 .
let us mention two of them .    1
.   random variable @xmath57 has exponential tail .
more precisely , @xmath78 for some positive constants @xmath79 and @xmath80 .
in this case , inequality ( [ eq4 ] ) is equivalent for sufficiently large @xmath77 to @xmath81 which is obviously true for large @xmath77 .
2 .   @xmath82 has power tail , that is @xmath83 , where @xmath84 in view of existence of finite second moment .
simple calculations show that ( [ eq4 ] ) is equivalent as @xmath85 to @xmath86 the last inequality is true for @xmath87 .
let us note that the function @xmath68 has a jump at zero .
however , one can obtain similar effect without such jump by using a smoothing procedure , that is by approximating @xmath68 by smooth functions .
`` put tail down '' procedure allows us to obtain more outliers in view of two its elements .
first element consists in changing the tail by smaller , but proportional to previous with coefficient @xmath88 .
the second element consist in moving a part of mass into origin ( or into a small neighborhood of it ) , which reduces the variance .
one of the arguments used to support the use of stable distributions in finance is that the observations may be considered as sums of a large number of random variables .
however , to have the convergence to stable distribution , the summands themselves must have heavy tails , which seems to be unnatural .
moreover , as we saw in sections [ s2 ] , [ s3 ] , the probability to observe large number of outliers is not in agreement with real data .
however , alternatives to gaussian distribution are not restricted by stable distributions only .
there are limit laws for the sums of i.i.d .
random variables for _ random number of summands_. the theory of such limit distributions has been developed by robbins , dobrushin , gnedenko and others .
the description of this theory can be found in the book @xcite .
for some new results , see @xcite . as shown in these publications , the sums of a random number of i.i.d .
random variables converges to so - called @xmath89-normal distribution .
the form of this distribution depends on the law for number of summands . in the case of geometric distribution for the number of summands the limit law is laplace distribution instead of gaussian . for the case of
transformed negative binomial distribution for the number of summands the role of gaussian law is played by symmetric gamma distribution .
many other examples are given in @xcite .
all such distributions have finite second moment .
for many of them , the probability ( [ eq2 ] ) is essentially higher than for gaussian law .
for example , laplace distribution gives the limit value of probability @xmath36 for @xmath37 equal to @xmath90 , versus @xmath44 for gaussian distribution .
our suggestion is that such distributions are good alternatives to gaussian and stable laws .
in the present paper , we discuss contra - arguments concerning the use of pareto - lev distributions for modeling in finance . 
 it appears that such probability laws do not provide sufficient number of outliers observed in real data . 
 connection with the classical limit theorem for heavy - tailed distributions with such type of models is also questionable . 
 the idea of alternative modeling is given .    
 * keywords*:outliers ; financial indexes ; heavy tails ; stable distributions