The interest of this paper is independent Student’s t increments. These increments are independent draws from a Student’s t-distribution with support, a truncated Student’s t -distribution with support, or an effectively truncated Student’s t-distribution with support, but which has a multiplicative envelope which effectively truncates the distribution. Here is the scale parameter for the Stu- dent’s t-distribution and is a real constant.

These independent Student’s t increments can be used to generate a random walk such as the Markov sequence, , , , where the are independent draws from a Student’s t-distribution, a truncated Student’s t-distribution, or an effectively truncated Student’s t-distribution.

Attention will be restricted to Student’s t-distributions with location parameter (i.e., mean), scale factor, and shape parameter, which cover the Cauchy distribution, for which, to the Gaussian or normal distribution, for which.

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 will represent random variables, and specific realizations of the random variables x and will be represented as x and. A stochastic process, which is a family of functions of time, is then whereas is a random variable for some constant and is a number in that both t and the value of are specified.

A Student’s t-distribution with location parameter, shape parameter, and scale parameter, is given by [1] - [3]

with. gives the probability that a random draw from the Student’s t-dis- tribution lies in the interval.

A truncated Student’s t-distribution with location parameter, shape parameter, and scale parameter, is given by

where the rectangle function if and for has been used to truncate the

distribution and limit support to.

A Student’s t-distribution is obtained from a mixture of a normal distribution with a standard deviation that is distributed as inverse chi with support [4] -[7] . Let, then a is distributed as chi, , and

Using chi as defined above and a normal distribution with zero mean and standard deviation of, the mixing integral when evaluated from to yields a Student’s t-distribution

with a mean of zero, shape parameter, and a scale parameter of.

The probability that, , is needed to normalize properly a truncated chi distribution. A left-truncated chi distribution is zero for values:

An effectively truncated Student’s t-distribution is the pdf for a mixture of a left-truncated chi and normal distribution:

is a Student’s t-distribution with shape parameter and scale parameter.

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 and 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 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 mo- ments 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 have continuous sample paths. Random walks are shown for independent increments drawn from a uniform distribution, from a normal distribution, and from and 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.

Let g and h be zero mean probability density functions (pdf’s) with variances and, and let be the convolution of g and h:

is also a zero mean pdf and hence the variance of is

where is the Fourier transform of. From the convolution theorem, , and

since g and h are zero-mean pdf’s:, , and

,. Thus

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, and will exist for pdf’s that are continuous or have finite dis- continuities [10] p. 9. The derivatives of the transforms might not exist at some values of s owing to higher- order 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. However, provided that, derivatives at exist for the convolution

which is the Fourier transform of the truncated distribution,

where if and for.

The convolution of Equation (13) does not appear to have an analytic expression except at.

An expression for the convolution, Equation (13), can be written for as

from which the derivatives at can, with some effort, be calculated. For, and

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, a continuous function of s and for which derivatives exist at. This stands in stark contrast to the Fourier transform for the not-truncated function (i.e., for), which is .

Figure 1 shows the effect of convolution on the Fourier transform of a Cauchy distribution. The Cauchy dis- tribution was truncated as indicated in Equation (14) with T = 100. The scale parameter of the Cauchy distri- bution of Equation (14) is. The truncation thus removes values that have magnitudes greater than times the scale factor. The probability of an observation with magnitude >50 is 0.002, i.e., , for the distribution of Equation (14). For a normally distributed random variable with mean and standard deviation,.

Figure 2 shows similar quantities as Figure 1 but with. The probability of an observation with magnitude >5000 is for the Cauchy distribution of Equation (14). In a “normal” world, . Truncation smooths the characteristic function and keeps moments finite.

The variance of an n-fold convolution, , is. If, then the variance for the n-fold self-convolution is. The pdf for the sum of n-independent draws from the same parent distribution that is characterized by a pdf f with variance 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, , , , where the, are independent Student’s t increments, i.e., the are independent draws from a Student’s t-distribution. The sequence is homogeneous since the pdf’s for each are independent of n. The sequence is stationary since it is homogeneous and all have the

same pdf. Let be the time between increments. The total time t taken to acquire the sequence is. The mean of is zero and the variance of, , is

where is the variance of any of the Student’s t increments. Allow, which requires. The variance will remain finite and non-zero only if as, where is a constant. Thus varies linearly with sampling period and the variance of the sequence varies linearly with time t.

The linear dependence on time of the variance of the Markov sequence 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, which requires, results in a Wiener-Lévy process, which is a stochastic process that is continuous for almost all outcomes. Papoulis then shows is a normally distributed random variable. Papoulis assumed n samples were drawn from a binomial distribution

with 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-distri- butions 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 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. The fourth central moment,

, is proportional to the variance squared for a Student’s t-distribution (see below) and there-

fore is a constant. The variance of Q is then, which tends to zero linearly with as. Thus in a mean-square sense, the expectation of the quadratic variation is, i.e.,. The variance of the stochastic process increases linearly with time t. For Gaussian increments,. For Student’s t increments, is a simple function of the shape parameter, the scale parameter, 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 varies as. The scale factor for a normal distribution is, the standard deviation. For Brownian motion, is a normally distributed random variable and. For Brownian motion the increments are independent, Gaussian random variables.

For a Markov process, the sample paths are continuous functions of t, if for any,

uniformly in, and [13] , p. 46. 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 (), symmetric pdf’s

or equivalently, since for,

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 and mean z has continuous sample paths. For simplicity in no- tation, assume that. For this process with pdf given by

the limit

equals zero and the sample paths are continuous.

An expansion of Equation (38) about shows that the dominant term goes as:

and thus the limiting value as is zero.

For samples paths that are created as the sums of independent draws from a Student’s t-distribution with (i.e., 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 about

shows that the dominant term is and thus the limit is infinity as.

Sample paths for both normal distributions and Student’s t-distributions (Cauchy) have

as required for consistency [13] , p. 47.

For a process with Student’s t-distribution increments, the sample paths are continuous if the limit

is zero. Since the limit is not zero, a process with Student’s t-distribution increments does not have con- tinuous paths.

For a process with Student’s t-distribution increments, the sample paths are continuous if the limit

is zero.

An expansion about shows that the dominant term for the condition for continuous sample paths for a Student’s t-distribution, Equation (50), is as:

Processes with Student’s t-distributions increments with have continuous paths since the limit as. However, the fourth moments for Student’s t-distributions with do not exist. Thus it would not be possible to use the mean-square variance of the quadratic variation to prove convergence of the ex- pecation of the quadratic variation to. See Equation (22) and associated discussion. The moments exist for truncated and effectively truncated Student’s t-distributions.¹

Independent Student’s t increments, from which stochastic processes such as random walks are created, are investigated. Attention is restricted to increments from not-truncated, truncated, and effectively truncated Student’s t-distributions with shape parameters, which covers a broad range of distributions: from Cauchy distributions, for which, to normal or Gaussian distributions, for which. A Student’s t-distribution can be obtained as a mixture of a chi distribution of the reciprocal of and a normal distribution with standard deviation of. Effectively truncated t-distributions arise from left-truncation of the chi distri- butions in the mixing integrals. An effectively truncated Student’s t-distribution has a Gaussian envelope that imparts interesting properties to the effectively truncated Student’s t-distribution.

Random walks, specifically Markov sequences, , , , where the are independent Student’s t increments, are considered. The development in time of the scale parameter of the Student’s t-distributions (not-truncated, truncated, and effectively truncated t-distributions) is investigated. It is found for distributions for which the variance exists that where is a constant and t is time. The variance exists for truncated and effectively truncated Student’s t-distributions, and for Student’s t-distributions with shape parameter. The development in time of the scale parameter for Student’s t-distributions is consistent with a normal distribution, for which the variance, which equals, in- creases linearly with time. A Gaussian (or normal) distribution is stable under convolution; in general, a Student’s t-distribution is not stable under convolution.

The continuity of the sample paths is investigated and it is found that truncated and effectively truncated Student’s t-distributions, and that Student’s t-distributions with, have continuous sample paths. This opens the possibility for modelling with a greater number of distributions.

Gardiner [13] , p. 79 defines a Wiener process as

with the constraints that, that is a continuous function of time t, and that is a Markov process. The requirement is not restrictive as any non-zero mean value of can be considered to be signal. Gardner explains that one normally assumes that is Gaussian and that. The assumption of Gaussian statistics follows from a desire to have continuous paths for. The white noise property of the Wiener process, i.e., , follows not from the assumption of Gaussian statistics for but from the assumption that is a Markov process. For a Markov process, is not determined probabilistically by any past values [13] , p. 78 and thus and are independent for all and.

A random walk process that is constructed from truncated or effectively truncated Student’s t increments with is continuous. This process can also be constructed under the Markov assumption that, where for simplicity increments are assumed to be draws from zero mean dis- tributions such that. Student’s t-distributions with appear also to be continuous, without the need for truncation. Thus it appears that there exists more than independent Gaussian increments for construction of random walks with continuous sample paths. Given continuous sample paths and second mo- ments that depend linearly on time for random walks with independent, not-truncated, truncated, and effectively truncated Student’s t-increments, it seems reasonable to speculate that the diffusion coefficients [13] [14] , p. 79, p. 133

exist and thus it should be possible to model noise in Langevin equations with appropriate t-distributions.

This work was funded by the Natural Science and Engineering Research Council (NSERC) Canada.

Daniel T.Cassidy, (2016) Student’s t Increments. Open Journal of Statistics,06,156-171. doi: 10.4236/ojs.2016.61014