Convergence rate to a lower tail dependence coefficient of a skew-t distribution

We examine the rate of decay to the limit of the tail dependence coefficient of a bivariate skew t distribution which always displays asymptotic tail dependence. It contains as a special case the usual bivariate symmetric t distribution, and hence is an appropriate (skew) extension. The rate is asymptotically power-law. The second-order structure of the univariate quantile function for such a skew-t distribution is a central issue.Comment: 14 page

Explicit forms for ergodicity coefficients of stochastic matrices

AbstractMotivated by explicit expressions appearing in the work of A. Rhodius (1993) for n×n stochastic matrices P, it is shown that ordinary matrix norms on Rn−1 for (n−1)×(n−1) matrices of the form APB can be used to generate results of this kind

Tail asymptotics for the bivariate skew normal in the general case

The present paper is a sequel to and generalization of Fung and Seneta (2016) whose main result gives the asymptotic behaviour as $u \to 0^{+}$ of $\lambda_L(u) = P(X_1 \leq F_1^{-1}(u) | X_2 \leq F_2^{-1}(u)),$ when $\bf{X} \sim SN_2(\boldsymbol{\alpha}, R)$ with $\alpha_1 = \alpha_2 = \alpha,$ that is: for the bivariate skew normal distribution in the equi-skew case, where $R$ is the correlation matrix, with off-diagonal entries $\rho,$ and $F_i(x), i=1,2$ are the marginal cdf's of $\textbf{X}$. A paper of Beranger et al. (2017) enunciates an upper-tail version which does not contain the constraint $\alpha_1=\alpha_2= \alpha$ but requires the constraint $0 <\rho <1$ in particular. The proof, in their Appendix A.3, is very condensed. When translated to the lower tail setting of Fung and Seneta (2016), we find that when $\alpha_1=\alpha_2= \alpha$ the exponents of $u$ in the regularly varying function asymptotic expressions do agree, but the slowly varying components, always of asymptotic form $const (-\log u)^{\tau}$, are not asymptotically equivalent. Our general approach encompasses the case $-1 <\rho < 0$, and covers all possibilities.Comment: 82 page

Chapitre 10 : entretien avec Eugene Seneta

  Eugene Seneta [[Seneta]] – professeur émérite à l’université de Sydney (School of Mathematics and Statistics) – est réputé pour ses contributions en probabilités et statistiques dont certaines ont débouché sur des applications aux domaines de la finance. Membre de l’Australian Academy of Sciences depuis 1985, il a aussi beaucoup contribué à l’histoire des probabilités et statistiques ; il revient dans cet entretien sur ses collaborations avec François Jongmans ainsi qu’avec Henri Breny, Be..

Topics in the theory and applications of Markov chains

The dissertation which follows is concerned with various aspects of behaviour within a set of states, J, of a discrete-time Markov chain, {Xn }, on a denumerable state space, S. A basic assumption with regard to J is that escape from any state of J into S-J may occur in a finite number of steps with positive probability. Since we are concerned only with behaviour within J, we may in general take J = {1,2,3,...} and represent S-J as a single absorbing state {0}. Thus without loss of generality S = {0,1,2,,..} with the states 1,2,3,... being transient. In addition, we frequently assume in the sequel that J is a single irreducible (i.e. intercommunicating) class, and sometimes that this class is aperiodic, these assumptions corresponding to the situations of greatest theoretical and practical importance. The subject matter which we treat falls naturally into two parts, according to which the thesis is divided. The aim of Part One is to develop results and techniques applicable to a wide class of problems, under general assumptions. This is done in the first three chapters, in which specific chains enter only as examples. On the other hand, specialized techniques are often applicable to specific chains of wide interest, such as certain models in genetics. This is particularly true of the Galton-Watson process, which is the subject of the following three chapters (Part Two) of the thesis. Three distinct but related aspects of transient behaviour within J are studied in the first part, each corresponding to a chapter

Relative entropy under mappings by stochastic matrices

AbstractThe relative g-entropy of two finite, discrete probability distributions x = (x1,…,xn) and y = (y1,…,yn) is defined as Hg(x,y) = Σkxkg (yk/kk - 1), where g:(-1,∞)→R is convex and g(0) = 0. When g(t) = -log(1 + t), then Hg(x,y) = Σkxklog(xk/yk), the usual relative entropy. Let Pn = {x ∈ Rn : σixi = 1, xi > 0 ∀i}. Our major results is that, for any m × n column-stochastic matrix A, the contraction coefficient defined as ηğ(A) = sup{Hg(Ax,Ay)/Hg(x,y) : x,y ∈ Pn, x ≠ y} satisfies ηg(A) ⩽1 - α(A), where α(A) = minj,kΣi min(aij, aik) is Dobrushin's coefficient of ergodicity. Consequently, ηg(A) < 1 if and only if A is scrambling. Upper and lower bounds on αg(A) are established. Analogous results hold for Markov chains in continuous time