San José State University
Department of Economics |
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applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
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and Their Characteristic Functions |
The analysis of stochastic processes involving variables with Lévy stable distribution has become more common-place. The formula for the characteristic function of such stable distributions is well known but the mathematical analysis involved in its derivation and proof is not so readily available. Such analyses and proofs are available but they are in monographs which are so old that libraries relegate them to their storage facilities. The purpose of this page is to present some of that mathematical analysis.
The theory of stable distribution was worked out with a fairly elaborate mathematical structure. One key element of that structure is the theory of infinitely divisible random variables. A random variable z would be said to be divisible if it could be represented as the sum of two independent random variables with identically distributions; i.e.,
where the probability distributions of z_{1} and z_{2} are identical. A variable z is said to be infinitely divisible if all values of n from 1 up it can be represented as the sum of n independent, identically distributed randome variables.
The condition for infinite divisibility can be more easily expressed in terms of characteristic functions. If Φ(ω) is the characteristic function of the distribution of z and Φ_{n}(ω) is the characteristic function of the common distribution of the n summands in
then
This means that
The Φ_{n}(ω) must be a valid characteristic function; i.e., it must be derived from a valid probabilitity distribution. There are certain properties that a characteristic function must have, such as that Φ(0) = 1 and that it must be continuous but these are not enough to guarantee that a function is the characteristic function of a valid probability distribution, one that has non-negative values for all values of the random variable.
Before showing that there are some characteristic functions that are not infinitely divisible it is perhaps appropriate to show that there are some that are. Consider the normal distributions. The characteristic function for a normal variable is of the form
where μ is the mean value and σ is the standard deviation.
The characteristic function for the identically distributed n summands of a normal variable is then:
The last line is the characteristic function of a normal variable with a mean of μ/n and standard deviation of σ/√n. This holds true for all and n and thus a normal variable is infinitely divisible.
To see that not all random variables are infinitely divisible one does not have to look any further than a discrete random variable that can take on only two distinct values, say b_{1} and b_{2}. For this variable to be infinitely divisible it would at least have to be divisible; i.e., representable as the sum of two independent, identically distributed random variables. Suppose such a distribution existed. If the variable could take on only one values then there would be no way to obtain the two values of b_{1} and b_{2}. Suppose the variable could take on two values, say a_{1} and a_{2}. Then the possible sums of two such random variables are a_{1} and a_{1}, a_{1} and a_{2}, and a_{2} and a_{2}. This precludes the possibility that the sum can have exactly two values, b_{1} and b_{2}. Thus a random variable that can take on two and only two distinct values cannot be divisible, much less infinitely divisible.
For a random variable that can take on only three distinct values divisibility is also precluded but for a different reason from the above. Let a random have the allowed values of b_{1}, b_{2} and b_{3} with probabilities of p_{1}, p_{2}and p_{3}, respectively and where p_{1}+p_{2}+p_{3} = 1. If this variable were to be represented as the sum of two independent, identically distributed variables those variables could only take on two values, say a_{1} and a_{2}, with probabilities p and q, respectively. The possible values of the sum of the two variables are 2a_{1}, a_{1}+a_{2}, 2a_{2}. Values of a_{1} and a_{2} can be found such that:
if and only if b_{2} = (b_{1}+b_{3}). Suppose that is the case. The probabilities of 2a_{1}, a_{1}+a_{2}, and 2a_{2} are p^{2}, 2pq and q^{2}, respectively. The conditions
can be satisfied only if p_{2} = 2(p_{1}p_{3})^{1/2}. Any three-valued random variable for which this is not true cannot be divisible, much less infinitely divisible. Thus in general discrete valued random variables cannot be infinitely divisible.
Given the conditions to be satisfied it somewhat surprising that any random variable would be infinitely divisible, but we know from the previous that at least normal variables are infinitely divisible. Actually many of the more statistically significant types of random variables are infinitely divisible.
The conditions for infinite divisibility can be equivalently (and more conveniently) expressed in terms of the log-characteristic functions; i.e.,
Some of the infinitely divisible types of random variables are
Some Infinitely Divisible Types of Random Variables | ||
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Type | Log-Characteristic Function | Divided Log-Characteristic Function |
Normal | iμω - (σ^{2}/2)ω^{2} | i(μ/n)ω - ((σ/√n)^{2}/2)ω^{2} |
Poisson | λ(e^{iω}-1) | (λ/n)(e^{iω}-1) |
Cauchy | iμω-a|ω| | i(μ/n)ω-(a/n)|ω| |
Kolmogorov's Formula
Formula of Lévy and Khintchine
(To be continued.)
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