ON A PROBABILISTIC ITERATED FACTOR METHOD

A REMARK ON THE DISTRIBUTION OF ADDITION CHAINS, 2025

We prove the prime obstruction principle and the sparsity law. These two are collective assertions that there cannot be many primes in an addition chain.

viXra, 2021

We suggest here a probabilistic approach that helps to address some classical questions and problems of Number Theory, like the Goldbach Conjecture [1], distributions of twin-and primes and primes among arithmetic sequences and many others.

2006

We prove a law of the iterated logarithm for the Kolmogorov-Smirnov statistic, or equivalently, the discrepancy of sequences (nk!) mod,1. Here (nk) is a sequence of integers satisfying a sub- Hadamard growth condition and such that linear Diophantine equations in the variables nk do not have too many solutions. The proof depends on a martingale embedding,of the empirical process;

Probability Theory and Related Fields, 1976

Nagoya Mathematical Journal, 1986

Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.

Theoretical Computer Science, 1997

The performance attributes of a broad class of randomised algorithms can be described by a recurrence relation of the form T(r) = Q(X) + T(H(x)), where a is a function and H(x) is a random variable. For instance, T(x) may describe the running time of such an algorithm on a problem of size X. Then T(x) is a random variable, whose distribution depends on the distribution of H(x). To give high probability guarantees on the performance of such randomised algorithms, it suffices to obtain bounds on the tail of the distribution of T(x). Karp derived tight bounds on this tail distribution, when the distribution of H(x) satisfies certain restrictions. In this paper, we give a simple proof of bounds similar to that of Karp using standard tools from elementary probability theory, such as Markov's inequality, stochastic dominance and a variant of Chemoff bounds applicable to unbounded geometrically distributed variables. Further, we extend the results, showing that similar bounds hold under weaker restrictions on H(n). As an application, we derive performance bounds for an interesting class of algorithms that was outside the scope of the previous results.

SIAM Journal on Discrete Mathematics, 2004

In this paper we study the following question posed by H. S. Wilf: What is, asymptotically as n ! 1, the probablility that a randomly chosen part size in a random composition of an integer n has multiplicity m ? More speci cally, given positive integers n and m, suppose that a composition of n is selected uniformly at random and then, out of the set of part sizes in , a part size j is chosen uniformly at random. Let P(A (m) n) be the probability that j has multiplicity m. We show that for xed m, P(A (m) n) goes to 0 at the rate 1= ln n. A more careful analysis uncovers an unexpected result: (ln n)P(A (m) n) does not have a limit but instead oscillates around the value 1=m as n ! 1. This work is a counterpart of a recent paper of Corteel, Pittel, Savage and Wilf who studied the same problem in the case of partitions rather than compositions. Integer partitions (as deterministic objects) have been studied for quite some time, but Erd os and Lehner 5] were apparently the rst to study integer partitions from the probabilistic perspective, namely, they considered the set of all partitions, P(n), of an integer n, as a probability space equipped with the uniform probability measure. Quantities of interest are treated as random variables and one can study their probabilistic properties, most typically, the limiting properties as n ! 1. Erd os and Lehner, for example, considered the limiting distribution of the total number of parts in a partition. Their paper opened a new line of investigation. Goh and Schmutz 10] obtained the central limit theorem for the number of di erent part sizes in a random partition, that is, they proved that the number of di erent part sizes, appropriately normalized, has, approximately, the standard Gaussian distribution. (Several years earlier Wilf 17] found an asymptotic formula for the expected number of distinct part sizes.) This approach culminated in an important paper by Fristedt 9], who proved that the joint distribution of the multiplicities of part sizes is that of independent geometric random variables (Y k), with parameters (1 ? p k), conditioned on the event f P kY k = ng. Fristedt's work, in turn, opened new possibilities and resulted in further progress in our understanding of the structure of random partitions. A good example is a paper of Pittel 16] substantiating two well{known conjectures concerning integer partitions. Utilizing Fristedt's result, quite recently Corteel, Pittel, Savage and Wilf 3] provided an answer to the following question. Consider the following two{step sampling procedure: rst choose uniformly at random a partition of n. Then, out of all di erent part sizes in pick one uniformly at random. What is the asymptotic unconditional probability that this part size has a certain speci ed multiplicity, say, m? For example, partition = f3; 2; 2; 1; 1; 1g of the number 10 has three di erent part sizes 1, 2, and 3 and only one of them has multiplicity three, namely 1. Thus, for this particular partition, the probability of choosing a part that has multiplicity three is 1=3. In order to nd the unconditional probability of randomly choosing a part of multiplicity three in a randomly chosen partition of 10, one would have to average similar probabilities over all partitions of 10. Corteel, Pittel, Savage and Wilf showed that in general the probability in question approaches 1=(m(m+1)) (in particular, the probability that the randomly chosen part size in a random partition is unrepeated approaches 1=2 as n ! 1.) Wilf then asked the same question for random compositions: what is the asymptotic value of the probability that a randomly chosen part size in a random composition of an integer n has multiplicity m ? Our aim here is to provide an answer as complete as we can. On the \ rst level" of precision the answer is simple: for every xed m this probability approaches zero. One would then like to know the rate of this convergence. We will show that the rate is 1= ln n. Speci cally, if A (m) n is the event that a randomly chosen part size in a random composition of n has multiplicity m then there exist constants c 1 (m) c 2 (m) such that c 1 (m) (ln n)P(A (m) n) c 2 (m), for n 2. The next natural step is to nd possibly tight bounds on c 1 (m) and c 2 (m),

Israel Journal of Mathematics, 2003

In this paper we find a second class of sequences of random numbers (xn)n~=l (the orbit of the ergodic adding machine) such that the corresponding sequences of zeros and ones l[0,y)(Xn) (n = 1, 2,..., N) satisfy Central Limit Theorems with extremely small standard deviation aN-O(lov/i-o~), instead of O(v/'N), as N-+ oo. Dedicated to Professor Benjamin Weiss on the occasion of his 60 th birthday. 1 1 f ~zNJ 2 (1.2) peRinf Jo f J£ ~n~:o dydz>2-Sl°g2N"

1997

Acta Mathematica Academiae Scientiarum Hungaricae, 1970