_{1}

^{*}

Some moments and limiting properties of independent Student’s t increments are studied. Inde-pendent Student’s t increments are independent draws from not-truncated, truncated, and effectively truncated Student’s t-distributions with shape parameters 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 Student’s t-distributions are also continuous. Student’s <i>t</i> increments should thus be useful in construction of stochastic processes and as noise driving terms in Langevin equations.

The interest of this paper is independent Student’s t increments. These increments are independent draws from a Student’s t-distribution with support

These independent Student’s t increments can be used to generate a random walk such as the Markov sequence

Attention will be restricted to Student’s t-distributions with location parameter (i.e., mean)

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

A Student’s t-distribution with location parameter

with

A truncated Student’s t-distribution

where the rectangle function

distribution and limit support to

A Student’s t-distribution is obtained from a mixture of a normal distribution with a standard deviation

Using chi as defined above and a normal distribution with zero mean and standard deviation of

with a mean of zero, shape parameter

The probability that

An effectively truncated Student’s t-distribution

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

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

Section 4 is a conclusion.

Let g and h be zero mean probability density functions (pdf’s) with variances

where

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

example, consider

distribution. The derivatives in the transform domain do not exist at

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

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

from which the derivatives at

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

The variance

Following Papoulis [

same pdf. Let

where

The linear dependence on time of the variance of the Markov sequence

Papoulis [

with

Not all functions tend to a normal pdf under repeated convolution [

The dependence on time of the variance for the Markov sequence

and the mean-square limit of the variance of the quadratic variation can be used to show convergence.

The variance of Q is

fore

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

The

Closed form expressions for the second, fourth, and sixth central moment are given, along with the values of

The second central moment

The fourth central moment

The sixth central moment

Not all central moments exist when the region of support for the t-distribution is

Truncation of Student’s t-distributions keeps the moments finite and defined [

The integrals that define the truncated central moment for the

Closed form expressions for the central moments for a truncated

For truncated

the fourth central moment

and the sixth central moment

All of these moments are defined with the single restriction that

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

uniformly in

The condition for continuous sample paths, Equation (32), can be written in different forms. For independent, zero mean (

or equivalently, since

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

the limit

equals zero and the sample paths are continuous.

An expansion of Equation (38) about

and thus the limiting value as

For samples paths that are created as the sums of independent draws from a Student’s t-distribution with

does not have continuous sample paths. The limit

does not equal zero. An expansion of

shows that the dominant term is

Sample paths for both normal distributions and

as required for consistency [

For a process with

is zero. Since the limit is not zero, a process with

For a process with

is zero.

An expansion about

Processes with Student’s t-distributions increments with ^{1}

Consider a truncated Cauchy with support

The variance for a truncated Cauchy with support

The condition for continuity is that the limit

equals zero for any

If

The pdf for a mixture of a left-truncated chi distribution for

The tails of the pdf decrease as

The condition for continuous sample paths for

which, owing to symmetry in

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

about

and the limit

for

Since probability is

and the sample paths for stochastic processes that are created by summing independent draws from effectively truncated

Distribution | Mean | Std Dev | Skewness | Kurtosis | Q | |
---|---|---|---|---|---|---|

Uniform −0.5 | −0.008 | 0.292 | 0.058 | 1.80 | 174.7 | 0.085 |

Normal | −0.025 | 1.023 | 0.008 | 2.96 | 2,142 | 1.046 |

0.346 | 63.47 | 30.06 | 1231 | 8,245,552 | 4,026 | |

−0.143 | 5.849 | −0.903 | 24.33 | 70,068 | 34.21 | |

−0.134 | 5.823 | −0.318 | 41.98 | 69,440 | 33.91 | |

−0.039 | 1.731 | 0.328 | 20.16 | 6,136 | 2.996 | |

−0.040 | 1.727 | 0.282 | 19.35 | 6,107 | 2.982 | |

−0.039 | 1.727 | 0.288 | 19.45 | 6,109 | 2.983 |

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

Distribution | Mean | Std Dev | Skewness | Kurtosis | Q | |
---|---|---|---|---|---|---|

Uniform −0.5 | 0 | 0.289 | 0 | 1.80 | 171 | 0.083 |

Normal | 0 | 1 | 0 | 3 | 2,048 | 1 |

0† | - | 0† | - | - | - | |

0 | 5.589 | 0 | 27.50 | 63,982 | 31.24 | |

0 | 5.617 | 0 | 52.28 | 64,624 | 31.55 | |

0 | 0† | - | 6,144 | 3 | ||

0 | 1.693 | 0 | 37.04 | 5,873 | 2.868 | |

0 | 1.702 | 0 | 56.14 | 5,932 | 2.896 |

The pdf for a mixture of a left-truncated chi distribution for

The left truncation of the chi distribution imparts a multiplicative Gaussian envelope that effectively truncates the underlying t distribution.

All walks shown in

Gaussian, Cauchy,

The data in

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

Random walks, specifically Markov sequences

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

Gardiner [

with the constraints that

A random walk process that is constructed from truncated or effectively truncated Student’s t increments

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