Krishna B. Athreya

Bootstrapping Markov chains : countable case

Abstract Let X be an irreducible, aperiodic and positive recurrent Markov chain with a countably infinite state space S and transition probability matrix P. Let π be the stationary probability and Tk be the first hitting time of a state k. Given a realization {xj;0⩽j⩽n} of {Xj;0⩽j⩽n}, let Pn be the maximum likelihood estimate of P. In this paper, the distribution of the naive bootstrap of the pivot √n(Pn–P) is shown to appropriate that of the pivot as n↦∞. The approach used is via a double array of Markov chains for which a weak law and a central limit theorem are established. Next, in order to estimate analogous quantities for the stationary probability and hitting time distribution, two different bootstrap methods are discussed.

Effective Strong Dimension in Algorithmic Information and Computational Complexity

The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems.

Strong law for the bootstrap

Let X1, X2, X3, … be i.i.d. r.v. with E|X1| < ∞, E X1 = μ. Given a realization X = (X1,X2,…) and integers n and m, construct Yn,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution P∗(Yn,i = Xj) = 1n for 1 ⩽ j ⩽ n. (P∗ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X1 are given to ensure the almost sure convergence of (1m)Σmi=1 Yn,i toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).

A note on strong mixing of ARMA processes

We establish that certain stationary autoregressive moving average (ARMA) processes are strong mixing.

Laws of large numbers for bootstrappedU-statistics

Abstract For the bootstrapped mean, a strong law of large numbers is obtained under the assumption of finiteness of the rth moment, for some r>1, and a weak law of large numbers is obtained under the finiteness of the first moment. The results are then extended to bootstrapped U-statistics under parallel conditions. Stochastic convergence of the jackknifed estimator of the variance of a bootstrapped U-statistic is proved. The asymptotic normality of the bootstrapped pivot and the bias of the bootstrapped U-statistic are indicated.

Confidence intervals for endpoints of a c.d.f. via bootstrap

Abstract We propose methods of constructing confidence intervals for endpoints of a distribution. Under a mild condition on the tail of distribution, asymptotically correct confidence intervals are derived by bootstrapping Weissmans (Comm. Statist. Theory Method A 10 (1981) 549–557) statistics. It is also shown that a modification of this method works for type II censored data.

On the Number of Latin Rectangles and Chromatic Polynomial of L(Kr,s)

Using Mobius inversion formula it is shown that the total number of Latin rectangles of a given order can be expressed in terms of Mobius function for the lattice of partitions of a set and the number of colourings of certain graphs. We prove the result in a very general form. In fact, we generalize the notions of Latin rectangles and colourings of graphs and prove a theorem in this general setting. An equivalent form of the theorem which is handy for calculation is given. Various special cases are considered. In particular, we obtain the chromatic polynomials of the line graphs of K 3, k and K 4, k or equivalently the total number of 3 × k and 4 × k Latin rectangles with entries from an n -set.

Preferential Attachment Random Graphs with General Weight Function

Start with graph G 0 ≡ {V 1, V 2} with one edge connecting the two vertices V 1, V 2. Now create a new vertex V 3 and attach it (i.e., add an edge) to V 1 or V 2 with equal probability. Set G 1 ≡ {V 1, V 2, V 3}. Let G n ≡ {V 1, V 2, . . . , V n+2} be the graph after n steps, n ≥ 0. For each i, 1 ≤ i ≤ n+2, let d n (i) be the number of vertices in G n to which V i is connected. Now create a new vertex V n+3 and attach it to V i in G n with probability proportional to w(d n (i)), 1 ≤ i ≤ n+2, where w(·) is a function from N ≡ {1, 2, 3, . . .} to (0,∞). In this paper, some results on behavior of the degree sequence {d n (i)} n≥1,i≥1 and the empirical distribution are derived. Our results indicate that the much discussed power-law growth of d n (i) and power law decay of hold essentially only when the weight function w(·) is asymptotically linear. For example, if w(x) = cx 2 then for i ≥ 1, lim n d n (i) exists and is finite with probability (w.p.) 1 and , and if w(x) = cx p, 1/2 < p < 1 then for i ≥ 1, d n (i) grows like (log n)q where q = (1 − p)−1. The main tool used in this paper is an embedding in continuous time of pure birth Markov chains.

Another conjugate family for the normal distribution

It is shown that the family of densities f(z) = czp exp([lambda]1z-1 + [lambda]2z), [lambda]1, [lambda]2 [greater-or-equal, slanted] o, - [infinity] 0, as marginals for the variance gives rise to a new conjugate family for the normal distribution. This family includes the normal gamma family and is minimal in an appropriate sense. This family is known as the generalized inverse Gaussian distribution.

Random Logistic Maps II. The Critical Case

AbstractLet (Xn)∞0 be a Markov chain with state space S=[0,1] generated by the iteration of i.i.d. random logistic maps, i.e., Xn+1=Cn+1Xn(1−Xn),n≥0, where (Cn)∞1 are i.i.d. random variables with values in [0, 4] and independent of X0. In the critical case, i.e., when E(log C1)=0, Athreya and Dai(2) have shown that Xn→P0. In this paper it is shown that if P(C1=1)<1 and E(log C1)=0 then(i) Xn does not go to zero with probability one (w.p.1) and in fact, there exists a 0<β<1 and a countable set ▵⊂(0,1) such that for all x∈A≔(0,1)∖▵, Px(Xn≥β for infinitely many n≥1)=1, where Px stands for the probability distribution of (Xn)∞0 with X0=x w.p.1. A is a closed set for (Xn)∞0.(ii) If γ is the supremum of the support of the distribution of C1, then for all x∈A (a)