Student ’ s t Increments

Some moments and limiting properties of independent Student’s t increments are studied. Independent Student’s t increments are independent draws from not-truncated, truncated, and effectively truncated Student’s t-distributions with shape parameters ν ≥ 1 and can be used to create random walks. It is found that sample paths created from truncated and effectively truncated Student’s t-distributions are continuous. Sample paths for ν ≥ 3 Student’s t-distributions are also continuous. Student’s t increments should thus be useful in construction of stochastic processes and as noise driving terms in Langevin equations.


Introduction: Student's t Increments
The interest of this paper is independent Student's t increments.These increments are independent draws from a Attention will be restricted to Student's t-distributions with location parameter (i.e., mean) 0 µ = , scale factor β < ∞ , and shape parameter 1 ν ≥ , which cover the Cauchy distribution, for which 1 ν = , to the Gaussian or normal distribution, for which ν = ∞ .
To distinguish between time t and and a realization of a random variable that is distributed as a Student's t-distribution, a bold face t will be used with the name of the distribution and a regular face t will represent time.The symbols x and ξ will represent random variables, and specific realizations of the random variables x and ξ will be represented as x and ξ .A stochastic process, which is a family of functions of time, is then ( ) t ξ whereas ( ) i t t = ξ is a random variable for i t some constant and ( ) i t t ξ = is a number in that both t and the value of ξ = ξ are specified.A Student's t-distribution with location parameter 0 µ = , shape parameter ν , and scale parameter β , is given by [1]- [3] ( where the rectangle function ( ) has been used to truncate the Using chi as defined above and a normal distribution with zero mean and standard deviation of 1 a σ = , the mixing integral when evaluated from 0 a = to a = +∞ yields a Student's t-distribution ( ) ( ) with a mean of zero, shape parameter ν , and a scale parameter of β .
The probability that q > x , ( ) , is needed to normalize properly a truncated chi distribution.A left-truncated chi distribution ( ) ; , ; , d .
An effectively truncated Student's t-distribution ( ) is the pdf for a mixture of a left-truncated chi and normal distribution: ; , e ; , , d .; , 2π is a Student's t-distribution with shape parameter ν and scale parameter β .
This paper is organized as follows.The development in time of the variance for the sum of independent draws from distributions is reviewed in Section 2. It is shown that truncation of a Student's t-distribution keeps the moments finite and thus variances add, even if the distributions are not stable under convolution.Gaussian and Cauchy distributions are stable under self-convolution.A Gaussian convolved with a Gaussian yields a Gaussian.Student's t-distributions other than 1 ν = and ν = ∞ distributions are not stable under self-convolution.The tails of the self-convolution of Student's t-distributions are "stable"; only the deep tails retain the characteristic 1 t ν + power-law dependence of the original t-distribution [6] [8] [9].However, the fact that the moments are finite and variances add under convolution allows the time development of the variance to be determined.Examples of smoothing of the characteristic function owing to truncation are given and examples of the moments of distributions are given.
The continuity of sample paths is discussed in Section 3. It is shown that truncated and effectively truncated Student's t-distributions have continuous sample paths.It is also shown that sample paths created by Student's t-distributions with 3 ν ≥ have continuous sample paths.Random walks are shown for independent increments drawn from a uniform distribution, from a normal distribution, and from 1 ν = and 3 ν = Student's t-distri- butions.The samples paths for the different distributions were all simulated from the same sequence of pseudo random numbers.This enables observation of the effects of different shape parameters and truncations on the random walks.
Section 4 is a conclusion.

Variances Add under Convolution
Let g and h be zero mean probability density functions (pdf's) with variances 2 g σ and 2 h σ , and let f g h = * be the convolution of g and h: ( ) f x is also a zero mean pdf and hence the variance of ( ) Fourier transform of ( ) From the convolution theorem, ( ) ( ) ( ) since g and h are zero-mean pdf's: ( ) ( ) and variances add under convolution.The argument holds even if the means for g and h are non-zero.The argument also holds for distributions that are stable or are not-stable under convolution, or for combinations of distributions that might not retain shape under the action of convolution.The Fourier transforms ( ) ( ) , F s G s , and ( ) H s will exist for pdf's that are continuous or have finite discontinuities [10] p. 9.The derivatives of the transforms might not exist at some values of s owing to higherorder discontinuities, but truncation of the pdf will smooth the transform and remove the discontinuities.For example, consider . This distribution in the x domain is a Cauchy distribution.The derivatives in the transform domain do not exist at 0 s = .However, provided that T < ∞ , derivatives at 0 s = exist for the convolution which is the Fourier transform of the truncated distribution, ( ) ( ) where ( ) The convolution of Equation ( 13) does not appear to have an analytic expression except at 0 s = .An expression for the convolution, Equation (13), can be written for 0 s ≥ as from which the derivatives at 0 s = can, with some effort, be calculated.For T < ∞ , ( ) ( ) ( ) ( ) The smoothing power of the convolution of Equation ( 13) can be observed if the sinc function is replaced by a unit area rectangle function with a similar width as the main lobe of the sinc function.Using this approximation for the sinc function, the convolution of Equation ( 13) becomes and can be evaluated to give ( ) which is, for T < ∞ , a continuous function of s and for which derivatives exist at 0 s = .This stands in stark contrast to the Fourier transform for the not-truncated function (i.e., for T = ∞ ), which is 1 shows the effect of convolution on the Fourier transform of a Cauchy distribution.The Cauchy distribution was truncated as indicated in Equation ( 14) with T = 100.The scale parameter of the Cauchy distribution of Equation ( 14) is ( ) − .The truncation thus removes values that have magnitudes greater than 100π times the scale factor.The probability of an observation with magnitude >50 is 0.002, i.e.,

{ }
Pr 50 0.002 x > = , for the distribution of Equation (14).For a normally distributed random variable with mean 0 µ = and standard deviation σ , Figure 2 shows similar quantities as Figure 1 but with 10000 T = . The probability of an observation with magnitude >5000 is The pdf for the sum of n-independent draws from the same parent distribution that is characterized by a pdf f with variance 2 f σ is the n-fold self-convolution of the parent pdf f and the variance of the sum of the n-independent draws is For a process that is the summation of samples that are periodically drawn from a parent population, the variance of the process would be proportional to time.
Following Papoulis [11] p. 292, consider a homogenous and stationary Markov sequence [11] p. 530 0 0 , where the , 0,1, ,  , are independent Student's t increments, i.e., the i x are independent draws from a Student's t-distribution.The sequence is homogeneous since the pdf's for each i x are independent of n.The sequence is stationary since it is homogeneous and all i x have the   The linear dependence on time of the variance of the Markov sequence n y arises not because of Gaussian properties, but because of the assumed independence of samples.The variance of a summation of independent samples is the sum of the variance of each sample, and thus the variance will increase linearly with the number of samples.If the samples are obtained by periodic sampling, then the variance will increase linearly with time.
Papoulis [11] p. 292 writes that the limit as n → ∞ , which requires 0 s T → , results in a Wiener-Lévy process, which is a stochastic process that is continuous for almost all outcomes.Papoulis then shows y ∞ is a normally distributed random variable.Papoulis assumed n samples were drawn from a binomial distribution with 1 2 p = and appealed to the DeMoivre-Laplace theorem to obtain a normal distribution in the limit that ( ) [11], p. 66.Not all functions tend to a normal pdf under repeated convolution [10], p. 186.A Cauchy distribution is probably the most noted distribution that does not follow the central limit theorem.Not all Student's t-distributions tend to a normal distribution.According to Bracewell [10], p. 190, only functions with finite area, finite mean, and finite second moments will tend to normal distributions under repeated convolution.For convolution of non-identical functions, Lyapunov's condition on the ratio of absolute moments to power of the variance must be satisfied.
The dependence on time of the variance for the Markov sequence n y can be obtained in a slightly different manner than the approach of Papoulis [11] and in a manner that does not specify the underlying pdf's.Following Shreve [12], p. 98, the expectation of the quadratic variation can be calculated and the mean-square limit of the variance of the quadratic variation can be used to show convergence.
The variance of Q is , is proportional to the variance squared for a Student's t-distribution (see below) and therefore α is a constant.The variance of Q is then ( ) .The variance of the stochastic process increases linearly with time t.For Gaussian increments, 3 α = .For Student's t increments, α is a simple function of the shape parameter ν , the scale parameter β , and the degree and form of truncation of the underlying pdf.
As the moments and continuity of a stochastic process are of interest, these topics are covered in the following sections.In the following, it is assumed on the strength of the arguments in this section and owing to the assumption of independent increments, that the scale factor β varies as t .The scale factor for a normal distribution is σ β = , the standard deviation.For Brownian motion, y y ∞ = is a normally distributed random variable and y o t σ σ = .For Brownian motion the increments are independent, Gaussian random variables.µ ν β , which is the variance 2 σ , is proportional to 2 β and is valid for 2 ν > :

Moments for Student's t-Distributions
The fourth central moment µ ν β is proportional to 4 β and is valid for 4 ν > : The sixth central moment µ ν β is proportional to 6 β and is valid for 6 ν > : Not all central moments exist when the region of support for the t-distribution is [ ] , −∞ +∞ .The integrals that define the truncated central moment for the = 1 ν

Moments for Truncated Student's t-Distributions
Student's t-distribution are ( ) ( ) Closed form expressions for the central moments for a truncated 1 ν = distribution are given.As might be expected, ( )  is proportional to n β with a constant of proportionality that is a function of b and n.
For truncated 1 ν = Student's t-distributions, the second central moment the fourth central moment and the sixth central moment All of these moments are defined with the single restriction that b < ∞ (i.e., that the 1 ν = distribution is truncated).Since the tails of distributions for 1 ν > decrease more rapidly than for a 1 ν = distribution, the central moments can be evaluated for all truncated Student's t-distributions with 1 ν ≥ .In this sense, the Cauchy distribution is a worst case.

Continuous Sample Paths
For a Markov process, the sample paths are continuous functions of t, if for any 0 >  , ( ) uniformly in , z t , and t is the pdf for the process and t is time.The condition for continuous sample paths, Equation (32), can be written in different forms.For independent, zero mean ( 0 z = ), symmetric pdf's ( ) ( ) Both forms will be used.A stochastic process that is created as the sums of independent draws from a normal distribution (i.e., Gaussian increments) with variance 1 t S ∆ = and mean z has continuous sample paths.For simplicity in no-tation, assume that 0 z = .For this process with pdf given by ( ) ) equals zero and the sample paths are continuous.An expansion of Equation ( 38) about S = ∞ shows that the dominant term goes as ( ) and thus the limiting value as S → ∞ is zero.
For samples paths that are created as the sums of independent draws from a Student's t-distribution with 1 ν = (i.e., 1 ν = Student's t increments), which is a Cauchy distribution, the pdf does not have continuous sample paths.The limit ( ) ( ) ( ) does not equal zero.An expansion of ( ) ( ) shows that the dominant term is S  and thus the limit is infinity as S → ∞ .Sample paths for both normal distributions and as required for consistency [13], p. 47.For a process with 2 ν = Student's t-distribution increments, the sample paths are continuous if the limit ( ) is zero.Since the limit is not zero, a process with 2 ν = Student's t-distribution increments does not have con- tinuous paths.
For a process with 3 ν = Student's t-distribution increments, the sample paths are continuous if the limit ( ) is given by Equation (29) since the mean is zero and truncation keeps the integral finite.The truncation need not be severe to obtain useful results; the variance diverges linearly with b.
The condition for continuity is that the limit , then the limit is zero and a process with truncated Cauchy increments should have a continuous path.However, it is not clear that the limit is zero for any 0 >  .The limit is zero when all the area of the pdf is enclosed by  for any 0 >  .In fact, the limit is zero only for  equal to "infinity", since the maximum value (i.e., "infinity") allowed is b S .Support for the truncated Cauchy distribution was taken as , b S b S   −   .The support was chosen to scale with the scale factor of the distribution so that the distribution was truncated to include the same fraction of the area of the not-truncated distribution, regardless of the choice of the scale factor 1 S .That is, the truncation was chosen such that the value of ( ) 1 Z b , which is defined by Equation (28), is independent of the scale factor.

Sample Paths: Effectively Truncated ν = 1 Distribution
The pdf for a mixture of a left-truncated chi distribution for , a q > , and a normal distribution is [6], [7] ( ) The tails of the pdf decrease as ( ) exp 2 q ξ − for non-zero q, where 1 q is the maximum value of σ that is included in the mixing integral.
The condition for continuous sample paths for ( ) which, owing to symmetry in ξ , is equivalent to ( ) The equation can be written as Consider the inequality An analytic expression for the integral of the upper bound of the inequality can be found.The dominant term in a series expansion for and the limit . The scaling o q q S = ensures that the truncation scales appropriately with S and thus keeps constant the area in the tails of the pdf that has been truncated.
Since probability is 0 ≥ , i.e., ( ) ξ ν β = = ≥ , and the limit as S → ∞ of the integral of the   There is little difference in shape between a truncated Student's t-distribution and an effectively truncated Student's t-distribution.From taking limits of the pdf, continuous sample paths were found for the effectively truncated 1 ν = distribution, yet a truncated 1 ν = distribution did not appear to have continuous sample paths (c.f.Equation (53) and related discussion).This discrepancy would seem to point to a problem with the con-dition for continuous sample paths or the interpretation of the condition for continuous sample paths.( ) Student's t-distribution with support [ ] , −∞ +∞ , a truncated Student's t -distribution with support [ ] truncated Student's t-distribution with support [ ] , −∞ +∞ , but which has a multiplicative ( ) 2 exp t − envelope which effectively truncates the distribution.Here β is the scale parameter for the Stu- dent's t-distribution and b < ∞ is a real constant.These independent Student's t increments can be used to generate a random walk such as the Markov sequence from a Student's t-distribution, a truncated Student's t-distribution, or an effectively truncated Student's t-distribution.
that a random draw ξ from the Student's t-dis- tribution lies in the interval d shape parameter ν , and scale parameter β , is given by s t-distribution is obtained from a mixture of a normal distribution with a standard deviation σ that is distributed as inverse chi with support [ ] distribution of Equation (14).In a "normal" world, the characteristic function and keeps moments finite.The variance 2 n σ of an n-fold convolution,

2 ,
for a Student's t-distribution with support [ ] for the second, fourth, and sixth central moment are given, along with the values of ν for which the expressions are valid.The second central moment ( ) Truncation of Student's t-distributions keeps the moments finite and defined[6] [7].As an example, consider a Student's t-distribution with one degree of freedom, = 1 ν , (i.e., a Cauchy or Lorentzian distribution) ∞ is a scale parameter and b is a number.Provided that b < ∞ , central moments for the truncated Cauchy (and for all other truncated Student's t-distributions with 1 ν ≥ ) exist.

1 3. 1 .
An expansion about S = ∞ shows that the dominant term for the condition for continuous sample paths for a 3 ν = Student's t-distribution, Equation (50), is 1 S as S → ∞ : 's t-distributions increments with 3 ν ≥ have continuous paths since the limit as 0 S → ∞ = .However, the fourth moments for Student's t-distributions with 4 ν ≤ do not exist.Thus it would not be possible to use the mean-square variance of the quadratic variation to prove convergence of the expecation of the quadratic variation { } E Q to 2 o t σ .See Equation (22) and associated discussion.The moments exist for truncated and effectively truncated Student's t-distributions.Sample Paths: Truncated Cauchy Consider a truncated Cauchy with support [ ] figure clearly shows that there is little difference between the truncated and effectively truncated 1 ν = pdf for x b < where b is the point of truncation.The tail of the truncated distribution falls off infinitely fast whereas the tails of the effectively truncated distribution fall off at the same rate as a Gaussian pdf.Since the random walk with Gaussian increments has continuous sample paths, one would expect an effectively truncated 1 ν = distribution to have continuous sample paths as the roll- off of the tails is similar.And since the tails of the truncated 1 ν = distribution roll-off faster than a Gaussian, one would expect a truncated 1 ν = walk to have continuous sample paths.The tails of the Cauchy distribution (i.e., a not truncated 1 ν = Student's t-distribution) do not roll off as fast as a Gaussian.A Cauchy random walk does not have continuous samples paths.Large steps in the Cauchy random walk are obvious in Figure 3.
Thus in a mean-square sense, the expectation of the quadratic variation is2 s T → .

Table 2
for sample and parent statistics of the distributions used to generate the figures.For a Cauchy distribution with 1

Table 2 .
Parent statistics for distributions with