Random Toeplitz matrices: The condition number under high stochastic dependence

Circulant Matrix ◽

Singular Value ◽

The Matrix ◽

Matrix Formula ◽

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix [Formula: see text] of dimension [Formula: see text] under the existence of the moment generating function of the random entries is [Formula: see text] with probability [Formula: see text] for any [Formula: see text], [Formula: see text]. Moreover, if the random entries only have the second moment, the condition number satisfies [Formula: see text] with probability [Formula: see text]. Also, we analyze the condition number of a random symmetric circulant matrix [Formula: see text]. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix [Formula: see text] we establish [Formula: see text], where [Formula: see text] is the minimum singular value of the matrix [Formula: see text]. The matrix [Formula: see text] is a random circulant matrix and [Formula: see text], where [Formula: see text] are deterministic matrices, [Formula: see text] indicates the conjugate transpose of [Formula: see text] and [Formula: see text] are random diagonal matrices. From random experiments, we conjecture that [Formula: see text] is well-conditioned if the moment generating function of the random entries of [Formula: see text] exists.

On the Power Series Expansion of the Moment Generating Function

The American Statistician ◽

10.2307/2683601 ◽

1974 ◽

Vol 28 (2) ◽

pp. 58

Author(s):

Roch Roy ◽

Yves Lepage ◽

Marc Moore

Keyword(s):

Power Series ◽

Series Expansion ◽

Generating Function ◽

Power Series Expansion ◽

On an admissibility problem involving the moment generating function of the lognormal distribution

Metrika ◽

10.1007/s001840200199 ◽

2003 ◽

Vol 57 (1) ◽

pp. 63-70 ◽

Cited By ~ 1

Author(s):

Leila Mohammadi

Keyword(s):

Generating Function ◽

Lognormal Distribution ◽

A Note on Using the Moment Generating Function to Teach the Laws of Large Numbers

The American Statistician ◽

10.2307/2684977 ◽

1993 ◽

Vol 47 (3) ◽

pp. 199 ◽

Cited By ~ 1

Author(s):

Lambert H. Koopmans

Keyword(s):

Generating Function ◽

Laws Of Large Numbers ◽

Large Numbers ◽

Problèmes de premier passage résolubles par la méthode de séparation des variables

Journal of Applied Probability ◽

10.1017/s0021900200102827 ◽

1995 ◽

Vol 32 (02) ◽

pp. 337-348

Author(s):

Mario Lefebvre

Keyword(s):

Brownian Motion ◽

Stochastic Processes ◽

Generating Function ◽

First Passage Times ◽

Standard Brownian Motion ◽

First Passage ◽

The Moment ◽

Passage Times

In this paper, bidimensional stochastic processes defined by ax(t) = y(t)dt and dy(t) = m(y)dt + [2v(y)]1/2 dW(t), where W(t) is a standard Brownian motion, are considered. In the first section, results are obtained that allow us to characterize the moment-generating function of first-passage times for processes of this type. In Sections 2 and 5, functions are computed, first by fixing the values of the infinitesimal parameters m(y) and v(y) then by the boundary of the stopping region.

Mellin’s Transform and Application to Some Time Series Models

ISRN Applied Mathematics ◽

10.1155/2014/976023 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

S. S. Appadoo ◽

A. Thavaneswaran ◽

S. Mandal

Keyword(s):

Time Series ◽

Generating Function ◽

Mellin Transform ◽

Fuzzy Numbers ◽

Time Series Models ◽

Form Expression ◽

The Moment ◽

Close Form ◽

Skewness And Kurtosis

This paper uses the Mellin transform to establish the means, variances, skewness, and kurtosis of fuzzy numbers and applied them to the random coefficient autoregressive (RCA) time series models. We also give a close form expression to the moment generating function related to fuzzy numbers. It is shown that the results of the proposed time series models are consistent with those of the conventional time series models and that the developed concepts are straightforward and easily implemented.

On a Dual Model with Barrier Strategy

Journal of Applied Mathematics ◽

10.1155/2012/343794 ◽

2012 ◽

Vol 2012 ◽

pp. 1-13 ◽

Cited By ~ 3

Author(s):

Yuzhen Wen ◽

Chuancun Yin

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Generating Function ◽

Risk Model ◽

Dual Model ◽

Barrier Strategy ◽

Dividend Payments ◽

We consider the dual of the generalized Erlang(n)risk model with a barrier dividend strategy. We derive integro-differential equations with boundary conditions satisfied by the expectation of the sum of discounted dividends until ruin and the moment-generating function of the discounted dividend payments until ruin, respectively. The results are illustrated by several examples.