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Combinations of Random Variables and Cumulative Sums of Random Disturbances |
The purpose of this material is to derive the characteristic function of a linear combination of random variables. The properties of the characteristic function of a probability distribution have been developed elsewhere. The characteristic function of a probability distribution is its Fourier transform. This results in the characteristic function of the sum of two random variables being the product of their characteristic functions. More conveniently this relationship is expressed in terms of the logarithms of the characteristic functions; i.e., the logarithm of the characteristic function of two random variables is the sum of the logarithms of their characteristic functions.
Let the probability density function (a.k.a. distribution function) of a random variable y(t) be denoted as p(y) and its characteristic function as φ(ω). Let Y(t)=by(t), where b may be positive or negative. The distribution function for Y(t), P(Y), is given by the general rule
Since the characteristic function for P(Y) is defined as
A change in the variable of integration from Y to y=Y/b results in
If φ(ω) is the characteristic function of p(y) then the above relation means that
To see the consequences of this rule consider the characteristic function for a normal distribution; i.e.,
where μ is the mean value of the variable and σ is its standard deviation. The symbol i stands for the square root of −1.
Thus
This means that for the Y distribution the mean value is the mean value for the y distribution multiplied by factor of b and likewise the standard deviation for the Y distribution is standard deviation for the y distribution multiplied by a factor of b. However the effect on the standard deviation is independent of the sign of b because the term is squared.
One significant implication of the above is that the probability distribution function of the negative of a variable is given by p(-y) and that its characteristic function is given by φ(-ω). Thus the characteristic function of z equal to the difference of random variable x and y is
For x and y with normal distributions with means μ_{x} and μ_{y} and standard deviations of σ_{x} and σ_{y}, respectively, this works out as
Let z=ax+by. Then
For normal distributions this works out as
Let z_{n}=x_{1}+x_{2}+…+x_{n} and let φ_{x}(ω) be the common characteristic function for the x_{i}'s. Then
For the x_{j}'s being normal with mean μ and standard deviation σ
Let Y(t)=∫_{0}^{t}u(t)dt where u(s) for all s is a random variable with a normal distribution have mean μ and standard deviation σ. Then
Now consider a moving average of the Y values given by
The integral ∫_{t-½k}^{t+½k}sds is equal to
This means that
Now consider a variable which is a unit time difference in the moving average; i.e.,
This variable represents the slope of the moving average function Z(t) and will have a normal distribution with
Let x(t) be a twice differentiable function of time. The curvature of this function is given by the second derivative x"(t). The average curvature over an interval of length L is given by
Now consider the variable W(t) defined as the difference in the slope of the moving average Z(t) over an interval of length L; i.e.,
This variable represents the average curvature of the function Z(t) over an interval of L. It has a normal distribution with the characteristic function being given by
Having the standard deviation of the average curvature being inversely dependent upon the product of the square root of the moving averaging interval k and the interval L is a powerful influence in clustering the average curvature values around the mean value of 0. The curvature of a straight line is zero. Thus the times series for Z(t) appearing to be linear. The moving average of random disturbances appears to have trends. The trends from a minimum to a maximum or from a maximum to minimum necessarily have an average curvature of zero. From the foregoing then these apparent trends will generally appear to be linear.
The temperature of any body, including the Earth's surface, is the accumulation of the net heat energy inputs to it. Here is the record of the average global temperatures.
There appears to be a series of linear trends. The foregoing explains why these trends apperar to be linear.
(To be continued.)
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