Size bias occurs famously in waiting-time paradoxes, undesirably in sampling schemes, and unexpectedly in connection with Stein's method, tightness, analysis of the lognormal distribution, Skorohod embedding, infinite divisibility, branching processes, and number theory. In this paper we review the basics and survey some of these unexpected connections.
According to a 1975 result of T. Kaijser, if some nonvanishing product of hidden Markov model (HMM) stepping matrices is subrectangular, and the underlying chain is aperiodic, the corresponding α-chain has a unique invariant limiting measure λ.Here the α-chain {αn} = {(αni)} is given byYn)} is a finite state HMM with unobserved Markov chain component {Xn} and observed output component {Yn}. This defines {αn} as a stochastic process taking values in the probability simplex. It is not hard to see that {αn} is itself a Markov chain. The stepping matrices M (y) = (M (y)ij) give the probability that (Xn, Yn) = (j, y), conditional on Xn−1 = i. A matrix is said to be subrectangular if the locations of its nonzero entries forms a cartesian product of a set of row indices and a set of column indices. Kaijser's result is based on an application of the Furstenberg-Kesten theory to the random matrix products M (Y1)M (Y2) · · · M (Yn). In this paper we prove a slightly stronger form of Kaijser's theorem with a simpler argument, exploiting the theory of e chains. . This reprint differs from the original in pagination and typographic detail. 1 2 F. KOCHMAN AND J. REEDSof substochastic matrices, the stepping matrices; and probability calculations with HMMs involve, at least conceptually, lengthy matrix products of stepping matrices. Accordingly, Kaijser's analysis utilized the Furstenburg-Kesten theory [6] of random matrix products. However, by exploiting the theory of e-chains, in particular Theorem 18.4.4 of [11], we are able to give a simpler proof of a result slightly stronger than that in [9]. Briefly, an HMM [2] consists of a pair of stochastic processes, {X n } and {Y n }, taking values in finite sets X and Y, such that {X n } is a Markov chain and each Y n is a probabilistic function of (X n−1 , X n ). In modeling applications [5,10], the "observable" marginal process {Y n } is "output" from the "hidden" process {X n }.The transition structure of (X n , Y n ) can be specified by the stepping matrices M (y) = (M (y) ij ), given by
Let f be an analytic germ on Cn+. Then there is an analytic linear partial differential operator P with poly nomial dependence on s, and a polynomial b(s), such that pfs l = b(s)fs. This paper contains a simple existence proof and geometric interpretation in the case when f has an isolated critical zero at the origin, and is contained in its Jacobian ideal of first partial derivatives. Letfk+ I = Xa1af/ax1. Then if we set x = Za O/oxj, we have xf = fk+ 1. Since x is a derivation, Xfs = fs+ k. We shall restrict to the special case that k = 0, so that xi = f. Then for any polynomial p(s), we have p(X)fs = p(s)fs. In lieu of showing here that (s + 1)b(s) is minimal, we shall make several remarks. Remarks: 1. A proof of minimality in this case can be based on the ideas of Malgrange (5) together with an explicit computation of the action of the Gauss-Manin connection on the vanishing cohomology of f.2. By a theorem of Saito (6), the condition that fE (af/axi) implies that f is equivalent, via an analytic change of coordinates, to a weighted homogeneous polynomial. In that case we may select x = XacxOa/ox1 for suitable aj E Q. Then it follows from the faithful flatness of C [[x]] over C [x]o that the construction described above yields an operator P with polynomial coefficients. Hence, the formula [1] is valid on all of Cn for weighted homogeneous f.3. In the case fE(af/axi), the operator Lx -1 can be identified with the residue operator of the Gauss-Manin connection. In the more general case that xf = fk+l, the operator f-k.f x -k -1 may be identified with the residue operator on Malgrange's space C/F (4).I am grateful to Victor Guillemin and John Hubbard for their assistance and encouragement in this work.
Asymptotics for Dickman's number theoretic function ρ(u), as u → ∞, were given de Bruijn and Alladi, and later in sharper form by Hildebrand and Tenenbaum. The perspective in these works is that of analytic number theory. However, the function ρ(•) also arises as a constant multiple of a certain probability density connected with a scale invariant Poisson process, and we observe that Dickman asymptotics can be interpreted as a Gaussian local limit theorem for the sum of arrivals in a tilted Poisson process, combined with untilting.In this paper we exploit and extend this reasoning to obtain analogous asymptotic formulas for a class of functions including, in addition to Dickman's function, the densities of random variables having Lévy measure with support contained in [0, 1], subject to mild regularity assumptions.
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