17 research outputs found
On bounded continuous solutions of the archetypal equation with rescaling
The `archetypal' equation with rescaling is given by
(), where is a probability measure; equivalently,
, with random and
denoting expectation. Examples include: (i) functional equation
; (ii) functional-differential
(`pantograph') equation (,
). Interpreting solutions as harmonic functions of the
associated Markov chain , we obtain Liouville-type results asserting
that any bounded continuous solution is constant. In particular, in the
`critical' case such a theorem holds subject to
uniform continuity of ; the latter is guaranteed under mild regularity
assumptions on , satisfied e.g.\ for the pantograph equation (ii). For
equation (i) with (, ), the
result can be proved without the uniform continuity assumption. The proofs
utilize the iterated equation (with a
suitable stopping time ) due to Doob's optional stopping theorem applied
to the martingale .Comment: Substantially revised. The title is modifie
Limit laws for sums of random exponentials
We study the limiting distribution of the sum S-N(t) = Sigma(i=1)(N) e(tXi) as t -> infinity, N -> infinity, where (X-i) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida's Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = -log P{X-I > x} is regularly varying at infinity with index 1 < rho < infinity. An appropriate scale for the growth of N relative to t is of the form e(lambda H0(t)), where the rate function Ho(t) is a certain asymptotic version of the cumulant. generating function H(t) = log E[e(tXi)] provided by Kasahara's exponential Tauberian theorem. We have found two critical points, 0 < lambda(1) < lambda(2) < infinity, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below lambda(2), we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent alpha = alpha(rho, lambda) ranging from 0 to 2 and with skewness parameter beta = 1. A limit theorem for the maximal value of the sample {e(tXi), i = 1,...,N} is also proved
Limit laws for sums of random exponentials
We study the limiting distribution of the sum S-N(t) = Sigma(i=1)(N) e(tXi) as t -> infinity, N -> infinity, where (X-i) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida's Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = -log P{X-I > x} is regularly varying at infinity with index 1 < rho < infinity. An appropriate scale for the growth of N relative to t is of the form e(lambda H0(t)), where the rate function Ho(t) is a certain asymptotic version of the cumulant. generating function H(t) = log E[e(tXi)] provided by Kasahara's exponential Tauberian theorem. We have found two critical points, 0 < lambda(1) < lambda(2) < infinity, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below lambda(2), we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent alpha = alpha(rho, lambda) ranging from 0 to 2 and with skewness parameter beta = 1. A limit theorem for the maximal value of the sample {e(tXi), i = 1,...,N} is also proved
Correlation effects in the trapping problem: general approach and rigorous results
The problem of Brownian survival among randomly located traps is considered with emphasis on the role of trap correlations. We proceed from the general representation of the survival probability as the expected value of the emptiness probability function applied to the Wiener sausage. Using the definition of (pure) trap attraction vs. repulsion in terms of the emptiness probability function, we prove the physical conjecture about the trapping slowdown or acceleration, according to the “sign” of correlations. Two specific models are studied along this line, in which the emptiness probability can be found explicitly; in particular, the long-time survival asymptotics is derived. A remarkable correlation effect of the survival probability dependence on the trap size in one dimension is also discussed