Dirichlet and quasi-Bernoulli laws for perpetuities

2012

Let X, B and Y be three Dirichlet, Bernoulli and beta independent random variables such that X ∼ D(a 0 ,. .. , a d), such that Pr(B = (0,. .. , 0, 1, 0,. .. , 0)) = a i /a with a = d i=0 a i and such that Y ∼ β(1, a). We prove that X ∼ X(1 − Y) + BY. This gives the stationary distribution of a simple Markov chain on a tetrahedron. We also extend this result to the case when B follows a quasi Bernoulli distribution B k (a 0 ,. .. , a d) on the tetrahedron and when Y ∼ β(k, a). We extend it even more generally to the case where X is a Dirichlet process and B is a quasi Bernoulli random probability. Finally the case where the integer k is replaced by a positive number c is considered when a 0 =. .. = a d = 1.

The relation of the Polya-Aeppli distribution of probabilities (also known as the Poisson-Geometric distribution) with random processes in the stationary case is considered in this article. Then the above mentioned distribution is derived, working with the probability generating functions, as limit of the distribution of rare events in a succession of Bernouilli trials with first order Markov dependence (that is, as a limit of Markov chains of rare events).

arXiv (Cornell University), 2014

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 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.

Journal of Theoretical Probability, 2008

We study probability distributions of convergent random series of a special structure, called perpetuities. By giving a new argument, we prove that such distributions are of pure type: degenerate, absolutely continuous, or continuously singular. We further provide necessary and sufficient criteria for the finiteness of p-moments, p > 0 as well as exponential moments. In particular, a formula for the abscissa of convergence of the moment generating function is provided. The results are illustrated with a number of examples at the end of the article.

Advances in Applied Probability, 2005

In this paper, we consider the asymptotic behavior of stationary probability vectors of Markov chains of GI/G/1 type. The generating function of the stationary probability vector is explicitly expressed by the R-measure. This expression of the generating function is more convenient for the asymptotic analysis than those in the literature. The RG-factorization of both the repeating row and the Wiener-Hopf equations for the boundary row are used to provide necessary spectral properties. The stationary probability vector of a Markov chain of GI/G/1 type is shown to be light tailed if the blocks of the repeating row and the blocks of the boundary row are light tailed. We derive two classes of explicit expression for the asymptotic behavior, the geometric tail, and the semigeometric tail, based on the repeating row, the boundary row, or the minimal positive solution of a crucial equation involved in the generating function, and discuss the singularity classes of the stationary probabilit...

Journal of Physics: Conference Series, 2021

We study a strongly Non-Markovian variant of random walk in which the probability of visiting a given site i is a function f of number of previous visits v(i) to the site. If the probability is inversely proportional to number of visits to the site, say f(i)=1/(v(i)+1)α the probability distribution of visited sites tends to be flat for α>0 compared to simple random walk. For f(i)=exp(-v(i)), we observe a distribution with two peaks. The origin is no longer the most probable site. The probability is maximum at site k(t) which increases in time. For f(i)=exp(-v(i)) and for α>0 the properties do not change as the walk ages. However, for α<0, the properties are similar to simple random walk asymptotically. We study lattice covering time for these functions. The lattice covering time scales as Nz, with z=2, for α ≤ 0, z>2 for a >0 and z<2 for f(i)=exp(-v(i)).

Communications in Statistics. Stochastic Models, 2000

Markov chains, whose transition matrices reveal a certain type of block-structure, find many applications in various areas. Examples include Markov chains of GI/M/1 type and M/G/1 type, and more generally Markov chains of Toeplitz type. Some Markov chains without a block-repeating structure can be also included; for example, level-dependent-quasi-birthand-death (LDQBD) processes. In analyzing this type of Markov chains, one may find that properties and/or probabilistic measures described or expressed by probability transition blocks from level to level often play a dominating role, while detailed transitions between states within the same level (block) are less important. In this paper, we introduce the concept of block-monitonicity and apply this notion to dealing with Markov chains possessing a block structure. A successful application in approximating stationary probability vectors of an infinite-state Markov chain is provided. We also hope that more applications of this concept can be exposed in the future.

Electronic Communications in Probability, 2014

Distributional findings are obtained relative to various quantities arising in Bernoulli arrays {X k,j , k ≥ 1, j = 1,. .. , r + 1}, where the rows (X k,1 ,. .. , X k,r+1) are independently distributed as Multinomial(1, p k,1 ,. .. , p k,r+1) for k ≥ 1 with the homogeneity across the first r columns assumption p k,1 = • • • = p k,r. The quantities of interest relate to the measure of the number of runs of length 2 and are S n = (Sn,1,. .. , Sn,r), S = limn→∞ S n , Tn = r j=1 Sn,j, and T = limn→∞ Tn, where Sn,j = n k=1 X k,j X k+1,j. With various known results applicable to the marginal distributions of the Sn,j's and to their limiting quantities Sj = limn→∞ Sn,j , we investigate joint distributions in the bivariate (r = 2) case and the distributions of their totals Tn and T for r ≥ 2. In the latter case, we derive a key relationship between multivariate problems and univariate (r = 1) problems opening up the path for several derivations and representations such as Poisson mixtures. In the former case, we obtain general expressions for the probability generating functions, the binomial moments and the probability mass functions through conditioning, an analysis of a resulting recursive system of equations, and again by exploiting connections with the univariate problem. More precisely, for cases where p k,j = 1 b+k for j = 1, 2 with b ≥ 1, we obtain explicit expressions for the probability generating function of S n , n ≥ 1, and S, as well as a Poisson mixture representation : S|(V1 = v1, V2 = v2) ∼ ind. Poisson(vi) with (V1, V2) ∼ Dirichlet(1, 1, b−1) which nicely captures both the marginal distributions and the dependence structure. From this, we derive the fact that S1|S1 + S2 = t is uniformly distributed on {0, 1,. .. , t} for all b ≥ 1. We conclude with yet another mixture representation for p k,j = 1 b+k for j = 1, 2 with b ≥ 1, where we show that S|α ∼ pα, α ∼ Beta(1, b) with pα a bivariate mass function with Poisson(α) marginals given by pα(s1, s2) = e −α α s 1 +s 2 (s 1 +s 2 +1)! (s1 + s2 + 1 − α) .

Journal of Theoretical Probability, 2000

Let {X k } k \ 1 be independent Bernoulli random variables with parameters p k . We study the distribution of the number or runs of length 2: that is S n =; n k=1 X k X k+1 . Let S=lim n Q . S n . For the particular case p k =1/(k+B), B being given, we show that the distribution of S is a Beta mixture of Poisson distributions. When B=0 this is a Poisson(1) distribution. For the particular case p k =p for all k we obtain the generating function of S n and the limiting distribution of S n for p=`lh+O(1/`n).

European Journal of Operational Research, 2011

This paper considers the subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains in two cases: (i) the phase transition matrix in non-boundary levels is stochastic; and (ii) it is strictly substochastic. For the case (i), we present a weaker sufficient condition for the subexponential asymptotics than those given in the literature. As for the case (ii), the subexponential asymptotics has not been studied, as far as we know. We show that the subexponential asymptotics in the case (ii) is different from that in the case (i). We also study the locally subexponential asymptotics of the stationary distributions in both cases (i) and (ii).