Probability Abstracts 57

1519. MOMENT INEQUALITIES FOR SYMMETRIC STATISTICS

R. Ibragimov and Sh. Sharakhmetov

In this paper, we prove analogues of Khintchine and Rosenthal's moment
inequalities for symmetric statistics (U-statistics) of arbitrary order. An
example that shows significance of each term in the analogues of Rosenthal's
bounds for symmetric statistics is constructed as well.

ibrag1r@mail.cmich.edu

• This article is available from the xxx mathematics archive as math.PR/0005004.

1520. AN INFORMATION-SPECTRUM APPROACH TO JOINT SOURCE-CHANNEL CODING

Te Sun Han

Given a general source $\sV=\{V^n\}\noi$ with {\em countably infinite} source
alphabet and a general channel $\sW=\{W^n\}\noi$ with arbitrary {\em abstract}
channel input and output alphabets, we study the joint source-channel coding
problem from the information-spectrum point of view. First, we generalize
Feinstein's lemma (direct part) and Verd\'u-Han's lemma (converse part) so as
to be applicable to the general joint source-channel coding problem. Based on
these lemmas, we establish a sufficient condition as well as a necessary
condition for the source $\sV$ to be reliably transmissible over the channel
$\sW$ with asymptotically vanishing probability of error. It is shown that our
sufficient condition coincides with the sufficient condition derived by Vembu,
Verd\'u and Steinberg, whereas our necessary condition is much stronger than
the necessary condition derived by them. Actually, our necessary condition
coincide with our sufficient condition if we disregard some asymptotically
vanishing terms appearing in those conditions. Also, it is shown that {\em
Separation Theorem} in the generalized sense always holds. In addition, we
demonstrate a sufficient condition as well as a necessary condition for the
$\vep$-transmissibility ($0\le \vep <1$). Finally, the separation theorem of
the traditional standard form is shown to hold for the class of sources and
channels that satisfy the (semi-) strong converse property.

han@toraja.hn.is.uec.ac.jp

• This article is available from the xxx mathematics archive as math.PR/0005058.

1521. THE IDENTIFICATION CAPACITY AND RESOLVABILITY OF CHANNELS WITH INPUT COST CONSTRAINT

Te Sun Han

Given a general channel, we first formulate the idetification capacity
problem as well as the resolvability problem with input cost constraint in as
the general form as possible, along with relevant fundamental theorems. Next,
we establish some mild sufficient condition for the key lemma linking the
identification capacity with the resolvability to hold for the continuous input
alphabet case with input cost constraint. Under this mild condition, it is
shown that we can reach the {\em continuous}-input fundamental theorem of the
same form as that for the fundamental theorem with {\em finite} input alphabet.
Finally, as important examples of this continuous-input fundamental theorem, we
show that the identification capacity as well as the resolvability coincides
with the channel capacity for stationary additive white (and also non-white)
Gaussian noise channels.

han@toraja.hn.is.uec.ac.jp

• This article is available from the xxx mathematics archive as math.PR/0005073.

1522. THE COVER TIME, THE BLANKET TIME, AND THE MATTHEWS BOUND

J. Kahn, J. H. Kim, L. Lovasz, V. H. Vu

The cover time C of a graph G is the expected time for a random walk starting
from the worst vertex to cover all vertices in G. Similarly, the blanket time B
is the expected time to visit all vertices within a constant factor of number
of times suggested by the stationary distribution. (Our definition will be
slightly stronger than this.) Obviously, all vertices are covered when the
graph is blanketed, and hence C <= B. The blanket time is introduced by Winkler
and Zuckerman motivated by applications in Markov estimation and distributed
computing. They conjectured B =O(C) and proved B=O(C ln n ). In this paper, we
introduce another parameter M motivated by Matthews' theorem and prove
  M/2 <= C <= B = O(( M ln ln n)^2).
  In particular, B = O(C (ln ln n)^2). The lower bound is still valid for the
cover time C(\pi) starting from the stationary distribution. We also show that
there is a polynomial time algorithm to approximate M within a factor of 2 and
so does for C within a factor of O((ln ln n)^2), improving previous bound of
O(ln n) of Matthews'.

vanhavu@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/0005121.

1523. LIMIT THEOREMS FOR HEIGHT FLUCTUATIONS IN A CLASS OF DISCRETE SPACE AND TIME GROWTH MODELS

Janko Gravner, Craig A. Tracy and Harold Widom

We introduce a class of one-dimensional discrete space-discrete time growth
models described by a height function $h_t(x)$ with corner initialization. The
growth rules depend upon two parameters: a probability $p$ and a positive
integer $\k$ that measures the refractory period. For $\k=1$ we prove, with one
exception, that the limiting distribution function of $h_t(x)$ (suitably
centered and normalized) equals a Fredholm determinant previously encountered
in random matrix theory. In particular, in the universal regime of large $x$
and large $t$ the limiting distribution is the Fredholm determinant with Airy
kernel. In the exceptional case, called the critical regime, the limiting
distribution seems not to have previously occurred. The proofs use the dual RSK
algorithm, Gessel's theorem, the Borodin-Okounkov identity and a novel,
rigorous saddle point analysis. In the fixed $x$, large $t$ regime, we find a
Brownian motion representation that in turn leads to a universality theorem
valid for all $\k$. Finally, for cases in which we have no proofs, we present
the results of simulations and formulate various conjectured limit theorems.

tracy@itd.ucdavis.edu

• This article is available from the xxx mathematics archive as math.PR/0005133.

1524. THE EMPIRICAL VALUES OF THE CRITICAL K-SAT EXPONENTS ARE WRONG

David B. Wilson

There has been much recent interest in the satisfiability of random Boolean
formulas. A random k-SAT formula is the conjunction of m random clauses, each
of which is the disjunction of k literals (a variable or its negation). It is
known that when the number of variables n is large, there is a sharp transition
from satisfiability to unsatisfiability; in the case of 2-SAT this happens when
m/n --> 1, for 3-SAT the critical ratio is thought to be m/n \approx 4.2. The
sharpness of this transition is characterized by a critical exponent, sometimes
called \nu=\nu_k. An article in the journal Science reported a detailed study
of \nu, in which it was estimated that \nu_3 = 1.5+-0.1, \nu_4 = 1.25+-0.05,
\nu_5 = 1.1+-0.05, \nu_6 = 1.05+-0.05, and \nu_k --> 1 as k --> infinity.
Similar estimates are given in a number of other articles as well. We give here
a simple proof that each of these exponents is at least 2 (provided the
exponent is well-defined). This result holds for each of the three standard
ensembles of random k-SAT formulas: m clauses selected uniformly at random
without replacement, m clauses selected uniformly at random with replacement,
and each clause selected with probability p independent of the other clauses.
We also obtain similar results for q-colorability and the appearance of a
q-core in a random graph.

dbwilson@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/0005136.

1525. STIELTJES INTEGRALS OF H\"OLDER CONTINUOUS FUNCTIONS WITH APPLICATIONS TO FRACTIONAL BROWNIAN MOTION

Anastasia Ruzmaikina

We give a new estimate on Stieltjes integrals of H\"older continuous
functions and use it to prove an existence-uniqueness theorem for solutions of
ordinary differential equations with H\"older continuous forcing. We construct
stochastic integrals with respect to fractional Brownian motion, and establish
sufficient conditions for its existence. We prove that stochastic differential
equations with fractional Brownian motion have a unique solution with
probability 1 in certain classes of H\"older-continuous functions. We give tail
estimates of the maximum of stochastic integrals from tail estimates of the
H\"older coefficient of fractional Brownian motion. In addition we apply the
techniques used for ordinary Brownian motion to construct stochastic integrals
of deterministic functions with respect to fractional Brownian motion and give
tail estimates of its maximum.

ar7f@weyl.math.virginia.edu

• This article is available from the xxx mathematics archive as math.PR/0005147.

1526. ON THE FINE STRUCTURE OF STATIONARY MEASURES IN SYSTEMS WHICH CONTRACT-ON-AVERAGE

Matthew Nicol, Nikita Sidorov, David Broomhead

Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We
form the iterated function system (IFS) by independently choosing the maps so
that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We
assume that the IFS contracts on average. We give an upper bound for the
Hausdorff dimension of the invariant measure induced on $\mathbb{R}^d$ and as a
corollary show that the measure will be singular if the modulus of the entropy
$\sum_i p_i \log p_i$ is less than $d$ times the modulus of the Lyapunov
exponent of the system. Using a version of Shannon's Theorem for random walks
on semigroups we improve this estimate and show that it is actually attainable
for certain cases of affine mappings of $\mathbb{R}$.

nikita.a.sidorov@umist.ac.uk

• This article is available from the xxx mathematics archive as math.PR/0005211.

1527. OCCUPATION TIME FLUCTUATIONS IN BRANCHING SYSTEMS

Don Dawson, L.G. Gorostiza, A. Wakolbinger

We consider particle systems in locally compact Abelian groups with particles
moving according to a process with symmetric stationary independent increments
and undergoing one and two levels of critical branching. We obtain long time
fluctuation limits for the occupation time process of the one-and two-level
systems. We give complete results for the case of finite variance branching,
where the fluctuation limits are Gaussian random fields, and partial results
for an example of infinite variance branching, where the fluctuation limits are
stable random fields. The asymptotics of the occupation time fluctuations are
determined by the Green potential operator G of the individual particle motion
and its powers $G^2, G^3$, and by the growth as $t\to\infty$ of the operator
$G_t=\int^t_0T_sds$ and its powers, where $T_t$ is the semigroup of the motion.
The results are illustrated with two examples of motions: the symmetric
$\alpha$-stable L\'evy process in $\erre^d$ $(0<\alpha\leq2)$,and the so called
c-hierarchical random walk in the hierarchical group of order N (0<c<N). We
show that the two motions have analogous asymptotics of $G_t$ and its powers
that depend on an order parameter $\gamma$ for their transience /recurrence
behavior. This parameter is $\gamma=d/\alpha-1$ for the $\alpha$-stable motion,
and $\gamma=\log c/\log (N/c)$ for the c-hierarchical random walk. As a
consequence of these analogies, the asymptotics of the occupation time
fluctuations of the corresponding branching particle systems are also
analogous. In the case of the c-hierarchical random walk, however, the growth
of $G_t$ and its powers is modulated by oscillations on a logarithmic time
scale.

ddawson@fields.utoronto.ca

• This article is available from the xxx mathematics archive as math.PR/0005231.

1528. SMOOTHNESS AND DECAY PROPERTIES OF THE LIMITING QUICKSORT DENSITY FUNCTION

James Allen Fill, Svante Janson 

Using Fourier analysis, we prove that the limiting distribution of the
standardized random number of comparisons used by Quicksort to sort an array of
n numbers has an everywhere positive and infinitely differentiable density f,
and that each derivative f^{(k)} enjoys superpolynomial decay at plus and minus
infinity. In particular, each f^{(k)} is bounded. Our method is sufficiently
computational to prove, for example, that f is bounded by 16.

jimfill@jhu.edu

• This article is available from the xxx mathematics archive as math.PR/0005235.

1529. A CHARACTERIZATION OF THE SET OF FIXED POINTS OF THE QUICKSORT TRANSFORMATION

James Allen Fill, Svante Janson 

The limiting distribution \mu of the normalized number of key comparisons
required by the Quicksort sorting algorithm is known to be the unique fixed
point of a certain distributional transformation T -- unique, that is, subject
to the constraints of zero mean and finite variance. We show that a
distribution is a fixed point of T if and only if it is the convolution of \mu
with a Cauchy distribution of arbitrary center and scale. In particular,
therefore, \mu is the unique fixed point of T having zero mean.

jimfill@jhu.edu

• This article is available from the xxx mathematics archive as math.PR/0005236.

1530. PERFECT SIMULATION FROM THE QUICKSORT LIMIT DISTRIBUTION

Luc Devroye, James Allen Fill,
 Ralph Neininger 

The weak limit of the normalized number of comparisons needed by the
Quicksort algorithm to sort n randomly permuted items is known to be determined
implicitly by a distributional fixed-point equation. We give an algorithm for
perfect random variate generation from this distribution.

jimfill@jhu.edu

• This article is available from the xxx mathematics archive as math.PR/0005237.

1531. MODELLING OF STOCK PRICE CHANGES: A REAL ANALYSIS APPROACH

Rimas Norvaisa

The paper discusses a path-wise approach to stock price modelling.

rimas@math.mit.edu

• This article is available from the xxx mathematics archive as math.PR/0005238.

1532. STOCHASTIC MONOTONICITY AND REALIZABLE MONOTONICITY

James Allen Fill, Motoya Machida 

We explore and relate two notions of monotonicity, stochastic and realizable,
for a system of probability measures on a common finite partially ordered set
(poset) S when the measures are indexed by another poset A. We give
counterexamples to show that the two notions are not always equivalent, but for
various large classes of S we also present conditions on the poset A that are
necessary and sufficient for equivalence. When A = S, the condition that the
cover graph of S have no cycles is necessary and sufficient for equivalence.
This case arises in comparing applicability of the perfect sampling algorithms
of Propp and Wilson and the first author of the present paper.

machida@mts.jhu.edu

• This article is available from the xxx mathematics archive as math.PR/0005267.

1533. DISTINGUISHED PROPERTIES OF THE GAMMA PROCESS AND RELATED TOPICS

N.Tsilevich, A.Vershik, M.Yor

We study fundamental properties of the gamma process and their relation to
various topics such as Poisson-Dirichlet measures and stable processes. We
prove the quasi-invariance of the gamma process with respect to a large group
of linear transformations. We also show that it is a renormalized limit of the
stable processes and has an equivalent sigma-finite measure (quasi-Lebesgue)
with important invariance properties. New properties of the gamma process can
be applied to the Poisson-Dirichlet measures. We also emphasize the deep
similarity between the gamma process and the Brownian motion. The connection of
the above topics makes more transparent some old and new facts about stable and
gamma processes, and the Poisson-Dirichlet measures.

natalia@pdmi.ras.ru

• This article is available from the xxx mathematics archive as math.PR/0005287.

1534. VALUES OF BROWNIAN INTERSECTION EXPONENTS III: TWO-SIDED EXPONENTS

Gregory F. Lawler, Oded Schramm, Wendelin Werner

This paper determines values of intersection exponents between packs of
planar Brownian motions in the half-plane and in the plane that were not
derived in our first two papers. For instance, it is proven that the exponent
$\xi (3,3)$ describing the asymptotic decay of the probability of
non-intersection between two packs of three independent planar Brownian motions
each is $(73-2 \sqrt {73}) / 12$. More generally, the values of $\xi (w_1,
>..., w_k)$ and $\tx (w_1', ..., w_k')$ are determined for all $ k \ge 2$,
$w_1, w_2\ge 1$, $w_3, ...,w_k\in[0,\infty)$ and all
$w_1',...,w_k'\in[0,\infty)$. The proof relies on the results derived in our
first two papers and applies the same general methods. We first find the
two-sided exponents for the stochastic Loewner evolution processes in a
half-plane, from which the Brownian intersection exponents are determined via a
universality argument.

schramm@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/0005294.

1535. ANALYTICITY OF INTERSECTION EXPONENTS FOR PLANAR BROWNIAN MOTION

Gregory F. Lawler, Oded Schramm, Wendelin Werner

We show that the intersection exponents for planar Brownian motions are
analytic. More precisely, let $B$ and $B'$ be independent planar Brownian
motions started from distinct points, and define the exponent $\xi (1,
\lambda)$ by $$ E[P[B[0,t] \cap B'[0,t] = \emptyset | B[0,t]]^\lambda] \approx
t^{-\xi(1, \lambda)/2}, t \to \infty. $$ Then the mapping $\lambda \mapsto \xi
(1, \lambda)$ is real analytic in $(0,\infty)$. The same result is proved for
the exponents $\xi (k, \lambda)$ where $k$ is a positive integer. In
combination with the determination of $\xi (k, \lambda)$ for integer $k \ge 1$
and real $\lambda \ge 1$ in our previous papers, this gives the value of $\xi
(k, \lambda)$ also for $\lambda \in (0,1)$ and the disconnection exponents
$\lim_{\lambda \searrow 0} \xi (k, \lambda)$. In particular, it shows that
$\lim_{\lambda \searrow 0} \xi(2, \lambda) = 2/3$ and concludes the proof of
the following result that had been conjectured by Mandelbrot: the Hausdorff
dimension of the outer boundary of $B[0,1]$ is 4/3 almost surely.

schramm@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/0005295.

1536. THE STANDARD DEVIATION EFFECT (OR WHY ONE SHOULD SIT FIRST BASE PLAYING BLACKJACK)

E. Munoz Garcia, R. Perez Marco

For a balanced cardcounting system we study the random variable of the true
count after a number of cards are removed from the remaining deck and we prove
a close formula for its standard deviation. As expected, the formula shows that
the standard deviation increases with the number of cards removed. This creates
a "standard deviation effect" with a two fold consequence: longer long run and
presumably larger fluctuations of the bankroll, but a small gain in playing
accuracy for the player sitting third base. The opposite happens for the player
sitting first base. Thus the optimal position in casino blackjack in terms of
shorter long run is first base.

ricardo@math.ucla.edu

• This article is available from the xxx mathematics archive as math.PR/0006017.

1537. GAUSSIAN LIMIT FOR DETERMINANTAL RANDOM POINT FIELDS

Alexander Soshnikov

We prove that under fairly general conditions properly rescaled determinantal
random point field converges to a generalized Gaussian random process.

soshniko@math.ucdavis.edu

• This article is available from the xxx mathematics archive as math.PR/0006037.

1538. MIXING TIMES FOR MARKOV CHAINS ON WREATH PRODUCTS AND RELATED HOMOGENEOUS SPACES

James Allen Fill, Clyde H. Schoolfield, Jr.
 

We develop a method for analyzing the mixing times for a quite general class
of Markov chains on the complete monomial group G \wr S_n (the wreath product
of a group G with the permutation group S_n) and a quite general class of
Markov chains on the homogeneous space (G \wr S_n) / (S_r \times S_{n - r}).
  We derive an exact formula for the L^2 distance in terms of the L^2 distances
to uniformity for closely related random walks on the symmetric groups S_j for
1 \leq j \leq n or for closely related Markov chains on the homogeneous spaces
S_{i + j} / (S_i \times S_j) for various values of i and j, respectively. Our
results are consistent with those previously known, but our method is
considerably simpler and more general.

jimfill@jhu.edu

• This article is available from the xxx mathematics archive as math.PR/0006076.

1539. RANDOM POLYNOMIALS HAVING FEW OR NO REAL ZEROS

Amir Dembo, Bjorn Poonen, Qi-Man Shao and Ofer Zeitouni

Consider a polynomial of large degree n whose coefficients are independent,
identically distributed, nondegenerate random variables having zero mean and
finite moments of all orders. We show that such a polynomial has exactly k real
zeros with probability n^{-b+o(1)}$ as n --> infinity through integers of the
same parity as the fixed integer k >= 0. In particular, the probability that a
random polynomial of large even degree n has no real zeros is n^{-b+o(1)}. The
finite, positive constant b is characterized via the centered, stationary
Gaussian process of correlation function sech(t/2). The value of b depends
neither on k nor upon the specific law of the coefficients. Under an extra
smoothness assumption about the law of the coefficients, with probability
n^{-b+o(1)} one may specify also the approximate locations of the k zeros on
the real line. The constant b is replaced by b/2 in case the i.i.d.
coefficients have a nonzero mean.

poonen@math.berkeley.edu

• This article is available from the xxx mathematics archive as math.PR/0006113.

1540. RANDOM WALKS ON WREATH PRODUCTS OF GROUPS

Clyde H. Schoolfield, Jr. 

We bound the rate of convergence to uniformity for certain random walks on
the complete monomial groups G \wr S_n for any group G. These results provide
rates of convergence for random walks on a number of groups of interest: the
hyperoctahedral group Z_2 \wr S_n, the generalized symmetric group Z_m \wr S_n,
and S_m \wr S_n. These results provide benchmarks to which many other random
walks, modeling a wide range of phenomena, may be compared using the comparison
technique, thereby yielding bounds on the rates of convergence to uniformity
for previously intractable random walks.

schoolfi@hustat.harvard.edu

• This article is available from the xxx mathematics archive as math.PR/0006118.

1541. A SIGNED GENERALIZATION OF THE BERNOULLI-LAPLACE DIFFUSION MODEL

Clyde H. Schoolfield, Jr. 

We bound the rate of convergence to stationarity for a signed generalization
of the Bernoulli-Laplace diffusion model; this signed generalization is a
Markov chain on the homogeneous space (Z_2 \wr S_n) / (S_r \times S_{n-r}).
Specifically, for r not too far from n/2, we determine that, to first order in
n, 1/4 n \log n steps are both necessary and sufficient for total variation
distance to become small. Moreover, for r not too far from n/2, we show that
our signed generalization also exhibits the ``cutoff phenomenon.''

schoolfi@hustat.harvard.edu

• This article is available from the xxx mathematics archive as math.PR/0006119.

1542. SEMIGROUPS, RINGS, AND MARKOV CHAINS

Kenneth S. Brown 

We analyze random walks on a class of semigroups called ``left-regular
bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon,
and Rockmore. Using methods of ring theory, we show that the transition
matrices are diagonalizable and we calculate the eigenvalues and
multiplicities. The methods lead to explicit formulas for the projections onto
the eigenspaces. As examples of these semigroup walks, we construct a random
walk on the maximal chains of any distributive lattice, as well as two random
walks associated with any matroid. The examples include a q-analogue of the
Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are
``generalized derangement numbers'', which may be of independent interest.

kbrown@math.cornell.edu

• This article is available from the xxx mathematics archive as math.PR/0006145.

1543. CHASING BALLS THROUGH MARTINGALE FIELDS

Michael Scheutzow and David Steinsaltz

We consider the way sets are dispersed by the action of
stochastic flows derived from martingale fields.  Under fairly
general continuity and ellipticity conditions, the following
dichotomy result is shown: any connected set either contracts to a
point under the action of the flow, or its diameter grows linearly
in time, with speed at least a positive deterministic constant $\gL$.
The linear growth may further be identified (again, almost
surely), with a much stronger behavior, which we call
``ball-chasing'': If $\psi$ is any path with Lipschitz constant
smaller than $\gL$, the ball of radius $\gep$ around $\psi(t)$
contains points of the image of $\X$ for an asymptotically positive fraction of
times $t$.  If the ball grows as the logarithm of time, there are
individual points in $\X$ whose images are eventually in
the ball.  This is, after a fashion, a converse to our earlier results
(with Michael Cranston) which showed that growth faster than linear is
impossible (a

• To see a preprint or other information provided by the author click here.
• Or here.
• Or here.

1544. RANDOM LOGISTIC MAPS AND LYAPUNOV EXPONENTS

David Steinsaltz

We prove that under certain basic regularity conditions, a random
iteration of logistic maps converges to a random point attractor
when the Lyapunov exponent is negative, and does not converge to a
point when the Lyapunov exponent is positive.

dstein@stat.berkeley.edu

• To see a preprint or other information provided by the author click here.

1545. SUBEXPONENTIAL LARGE DEVIATIONS FOR MARKOV CHAINS

David Steinsaltz

We derive general bounds for the deviations of partial sums along
the paths of a $V$-uniformly ergodic Markov chain.  These are more
general than previous results for uniformly ergodic Markov chains,
and allow the test function to be unbounded.  The tail bounds for
the probability that $\sum_{i=1}^k g(X_i)$ differs from
$k\pi(g)$ by more than $k^{1/2} t$ are not, in
general, exponential, but are of order $t^{-2n}$ as long as the
test function is bounded by $V^{1/2n}$.

dstein@stat.berkeley.edu

• To see a preprint or other information provided by the author click here.

1546. THE SPECTRAL GAP FOR A GLAUBER--TYPE DYNAMICS IN A CONTINUOUS GAS

Lorenzo Bertini, Nicoletta Cancrini and Filippo Cesi

We consider a continuous gas in a $d-$dimesional 
rectangular box with a finite range, positive 
pair potential,and we construct a Markov process in which 
particles appear and disappear with appropriate rates so 
that the process is reversible w.r.t. the Gibbs measure. If 
the thermodynamical paramenters are such that the Gibbs 
specification satisfies a certain mixing condition, then 
the spectral gap of the generator is strictly positive 
uniformly in the volume and boundary condition. The 
required mixing condition holds if, for instance, there is 
a convergent cluster expansion.

lorenzo@carpenter.mat.uniroma1.it  nicoletta.cancrini@roma1.infn.it  cesi@zephyrus.roma1.infn.it

• To see a preprint or other information provided by the author click here.

1547. LOCAL SUB-GAUSSIAN ESTIMATES ON GRAPHS, THE STRONGLY RECURRENT CASE

Andras Telcs

This paper proves upper and lower off-diagonal, 
sub-Gaussian transition probabilities estimates 
for strongly recurrent random walks under sufficient 
and necessary conditions. Several equivalent conditions 
are given showing the particular role of them and their 
influence to the connection between the sub-Gaussian 
estimates, parabolic and elliptic Harnack inequality.

h197tel@ella.hu

1548. VOLUME AND TIME DOUBLING OF GRAPHS AND RANDOM WALKS

Andras Telcs

This paper proves upper and lower off-diagonal, 
sub-Gaussian transition probability estimates 
for strongly recurrent random walks under sufficient 
and necessary conditions. Beside the known conditions 
- volume doubling and elliptic Harnack inequality, 
a new property - time doubling is introduced.

h197tel@ella.hu

1549. OPTIMAL STOPPING AND PERPETUAL OPTIONS FOR LEVY PROCESSES

Ernesto Mordecki

Solution to the optimal stopping problem for a Levy 
process and reward functions max(exp(x)-K,0) and 
max(K-exp(x),0), discounted at a constant rate is given in 
terms of the distribution of the overall supremum and 
infimum of the process killed at this rate.
Closed forms of this solutions are obtained under the 
condition of positive jumps mixed-exponentially distributed.
Results are interpreted as admissible pricing of perpetual 
American call and put options on a stock driven by a Levy 
process, and a Black-Scholes type formula is obtained.

mordecki@cmat.edu.uy

• To see a preprint or other information provided by the author click here.

1550. QUANTIFYING THE VALUE OF INITIAL INVESTMENT INFORMATION

Juergen Amendinger, Dirk Becherer, Martin Schweizer

We consider an investor maximizing his expected utility from terminal wealth
 with  portfolio decisions based on the  available information flow. 
This investor faces the opportunity to acquire some additional initial informati
on.
The subjective fair value of this  information for the  investor  is defined as 
 the amount of money  that he can pay  for it such that this cost is balanced
 out by the  informational advantage in terms of maximal expected utility.
We calculate this value for  common utility functions in the
setting of a complete market modeled by general semimartingales.
The main tools are results of independent interest, namely  a  martingale preser
ving change of measure and a martingale representation theorem for initially enla
rged filtrations.

becherer@math.tu-berlin.de

• To see a preprint or other information provided by the author click here.
• Or here.

1551. STRONG FELLER SOLUTIONS TO SPDE'S ARE STRONG FELLER IN THE WEAK TOPOLOGY

Bohdan Maslowski and Jan Seidler 

For a wide class of Markov processes on a Hilbert space $H$,
defined by semilinear stochastic partial differential 
equations, we show that their transition semigroups map 
bounded Borel functions to functions weakly continuous on 
bounded sets, provided they map bounded Borel functions into
functions continuous in the norm topology. In particular, 
an Ornstein-Uhlenbeck process in $H$ is strong Feller
in the norm topology if and only if it is strong Feller in 
the bounded weak topology. As a consequence, it is possible 
to strengthen results on the long-time behaviour of strongly 
Feller processes on $H$: we extend the embedded Markov 
chains method of constructing a $\sigma$-finite invariant 
measure by replacing recurrent compacts with recurrent balls, 
and in the transient case we prove that the last exit time 
from every weakly compact set is finite almost surely.

maslow@math.cas.cz seidler@math.cas.cz

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1552. IDENTIFICATION AND PROPERTIES OF REAL HARMONIZABLE FRACTIONAL LEVY MOTIONS.

Albert Benassi, Serge Cohen and Jacques Istas

In this article the class of Real Harmonizable Fractional L\'evy
Motions is introduced. It is shown that RHFLM share many properties
with Fractional Brownian Motion. These fields are locally
asymptotically self-similar with a constant index $ H,$ and have
Holderian paths.  Moreover  the identification of 
$H $ for RHFLM can be performed with the so-called generalized variation
method. Besides the Fractional Brownian Motion  this class contains
non-Gaussian fields that are  asymptotically self-similar at infinity
with a Real Harmonizable  Fractional Stable Motion of  index 
$\tilde{H} $ as tangent field. This last property should be useful to
model phenomena with multiscale behavior. 

scohen@cict.fr

1553. LONG-TIME TAILS IN THE PARABOLIC ANDERSON MODEL WITH BOUNDED POTENTIAL

Marek Biskup, Wolfgang Koenig

We consider the parabolic Anderson problem 
$\partial_t u=\kappa\Delta u+\xi u$ on 
$(0,\infty)\times \Z^d$ with random i.i.d. 
potential $\xi=(\xi(z))_{z\in\Z^d}$ and the 
initial condition $u(0,\cdot)\equiv1$. Our 
main assumption is that $\esssup\xi(0)=0$. 
In dependence of the thickness of the 
distribution $P(\xi(0)\in\cdot)$ close to its 
essential supremum, we identify both the 
asymptotics of the moments of $u(t,0)$ and 
the almost-sure asymptotics of $u(t,0)$ as 
$t\to\infty$ in terms of variational problems. 
As a by-product, we establish Lifshitz tails 
for the random Schroedinger operator 
$-\kappa\Delta-\xi$ at the bottom of its spectrum. 
In our class of $\xi$-distributions, the 
Lifshitz exponent ranges from $\frac d2$ 
to $\infty$; the power law is typically a
ccompanied by lower-order corrections.

koenig@math.tu-berlin.de

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1554. SCREENING EFFECT DUE TO HEAVY LOWER TAILS IN ONE-DIMENSIONAL PARABOLIC ANDERSON MODEL

Marek Biskup, Wolfgang Koenig

We consider the large-time behavior of 
the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ 
to the parabolic Anderson problem $\partial_t u=
\kappa\Delta u+\xi u$ with initial data $u(0,\cdot)=1$ 
and non-positive  finite i.i.d. potentials 
$(\xi(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, 
the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ 
is not determined solely by the upper tails of $\xi(0)$; 
too heavy lower tails of $\xi(0)$ accelerate the decay. 
The interpretation is that sites $x$ with large negative 
$\xi(x)$ hamper the mass flow and hence screen 
off the influence of more favorable regions of the 
potential. The phenomenon is unique to $d=1$. The 
result answers an open question from our previous study 
of this model in general dimension.

koenig@math.tu-berlin.de

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1555. TRANSITION DENSITY ASYMPTOTICS FOR SOME DIFFUSION PROCESSES WITH MULTI-FRACTAL STRUCTURES

Martin T. Barlow and Takashi Kumagai

We study the asymptotics as $t\to 0$ of the transition density of a class
of $\mu$-symmetric diffusions in the case when the measure $\mu$ has
a multi-fractal structure. These diffusions include singular time changes
of Brownian motion on the unit cube.

kumagai@i.kyoto-u.ac.jp

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