Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher–Snedecor random fields. Some simulation results are also presented.

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Sojourn measures of Student and Fisher-Snedecor random fields

Ergodicity And Stability Of Stochastic Processes

Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher-Snedecor random fields. Some simulation results are also presented.

This article offers a simplified approach to the distribution theory of randomly weighted averages or $P$-means $M_P(X):= \sum_{j} X_j P_j$, for a sequence of i.i.d.random variables $X, X_1, X_2, \ldots$, and independent random weights $P:= (P_j)$ with $P_j \ge 0$ and $\sum_{j} P_j = 1$. The collection of distributions of $M_P(X)$, indexed by distributions of $X$, is shown to encode Kingman's partition structure derived from $P$. For instance, if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}$, the $n$th moment of $M_P(X_p)$ is a polynomial function of $p$ which equals the probability generating function of the number $K_n$ of distinct values in a sample of size $n$ from $P$: $E (M_P(X_p))^n = E p^{K_n}$. This elementary identity illustrates a general moment formula for $P$-means in terms of the partition structure associated with random samples from $P$, first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters $(\alpha,\theta)$, found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an $(\alpha,\theta)$ mean. The analysis of these random means includes the characterization of $(0,\theta)$-means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of L\'evy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989).

Random weighted averages, partition structures and generalized arcsine laws

We consider a class of discrete-time Markov chains with state space [0, 1] and the following dynamics. At each time step, first the direction of the next transition is chosen at random with probability depending on the current location. Then the length of the jump is chosen independently as a random proportion of the distance to the respective end point of the unit interval, the distributions of the proportions being fixed for each of the two directions. Chains of that kind were the subjects of a number of studies and are of interest for some applications. Under simple broad conditions, we establish the ergodicity of such Markov chains and then derive closed-form expressions for the stationary densities of the chains when the proportions are beta distributed with the first parameter equal to 1. Examples demonstrating the range of stationary distributions for processes described by this model are given, and an application to a robot coverage algorithm is discussed.

On explicit form of the stationary distributions for a class of bounded Markov chains

We characterise the class of distributions of random stochastic matrices X with the property that the products X(n)X(n − 1)···X(1) of i.i.d. copies X(k) of X converge a.s. as n → ∞ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/27430B47035E27302D3E80D873119CBE/S0021900200011414a.pdf/div-class-title-a-characterisation-of-transient-random-walks-on-stochastic-matrices-with-dirichlet-distributed-limits-div.pdf

A characterisation of transient random walks on stochastic matrices with Dirichlet distributed limits

We characterise the class of distributions of random stochastic matrices $X$ with the property that the products $X(n)X(n-1) ... X(1)$ of i.i.d. copies $X(k)$ of $X$ converge a.s. as $n \rightarrow \infty$ and the limit is Dirichlet distributed. This extends a result by Chamayou and Letac (1994) and is illustrated by several examples that are of interest in applications.

Motivated by an approximation problem from mathematical finance, we analyse the stability of the boundary crossing probability for the multivariate Brownian motion process, with respect to small changes of the boundary. Under broad assumptions on the nature of the boundary, including the Lipschitz condition (in a Hausdorff-type metric) on its time cross-sections, we obtain an analogue of the Borovkov and Novikov (2005) upper bound for the difference between boundary hitting probabilities for "close boundaries" in the univariate case. We also obtained upper bounds for the first boundary crossing time densities.

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On approximation rates for boundary crossing probabilities for the multivariate Brownian motion process

The uniform law for sojourn times of processes with cyclically exchangeable increments is extended to the case of random fields, with general parameter sets, that possess a suitable invariance property.

Shaun McKinlay

Papers

A characterisation of transient random walks on stochastic matrices with Dirichlet distributed limits

On explicit form of the stationary distributions for a class of bounded Markov chains

A characterisation of transient random walks on stochastic matrices with Dirichlet distributed limits

On approximation rates for boundary crossing probabilities for the multivariate Brownian motion process

The Uniform Law for Sojourn Measures of Random Fields