Conference rusure::math

1812.0. "probability and stochastic processes newsletter" by STAR::ABBASI (only 35 days to graduation bash ...!) Fri Nov 05 1993 17:59

there is a distribution list for getting newsletter on probability and
    stochastic processess, if you want get it send mail to
    the address at the "from" below just saying to add you to this
    list. this is the first one i got today from them.
    
    \nasser
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From:	US2RMC::"[email protected]" "Probability Abstracts Account"  5-NOV-1993 17:52:55.14
To:	star::abbasi, [email protected]
CC:	
Subj:	PAS 17



Dear Colleague:                                               

In April 1991 Rich Bass, Mike Sharpe and I started the "Probability 
Abstract Service". The way it works is this: when you finish a paper, 
send me an abstract at [email protected].  About once every 
two months or so I e-mail out the abstracts that I have received 
since the last mailing. If anyone sees an abstract that interests
him or her, he or she contacts the author directly for a preprint.

The latest mailing follows. If you would like any of the older 
mailings, just let me know.

Regards
Chris Burdzy
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                                                            October 28, 1993
                                                            Letter 17
                                                            Abstracts 272-290


Dear Colleague:

If you have any abstracts for the next mailing, please send them to 
[email protected].  If you are not now on the mailing list 
and want to be added or if you would like copies of any of the previous 
mailings, send a message to the same address.

Regards
Rich Bass, Chris Burdzy, Mike Sharpe

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272. BLACK'S CONSOL RATE CONJECTURE

Derrell Duffie, Jin Ma, and Jiongmin Yong

This paper confirms a version of a conjecture by Fischer Black regarding
consol rate models for the term structure of interest rates. A consol rate
model is one in which the stochastic behavior of the short rate is influenced
by the consol rate. Since the consol rate is itself determined, via the
usual discouted present value formula, by the short rate, such models have
an inherent fixed point aspect.

Under an equivalent martingale measure, purely technical regularity conditions
are given for the stochastic differential equation defining the short rate
and the consol rate to be consistent with the definition of the consol rate
as the yield on a perpetual annuity. The results are based on an extension
of the recent developments in the theory of forward-backward stochastic
differential equations reported in Ma and Yong (1993) and Ma, Protter, and
Yong (1993) to infinite-horizon settings. Under additional compatibility
conditions, we also show that the consol rate is uniquely determined, and 
given as a function of the short rate.

[email protected]

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273. A UNIVERSAL CHUNG-TYPE LAW OF THE ITERATED LOGARITHM

Uwe Einmahl and David M. Mason

Let X(1), X(2), ... be iid random variables. We find sequences b(n) and
a(n) such that a universal Chung-type LIL holds, namely
lim inf max|S(k) - k b(n)|/a(n) is finite almost surely,
where S(k) denotes the sum of the first k of X(1),X(2),... .
If the underlying distribution is in the Feller class, we show that the lim
inf is positive with probability one.

[email protected] 

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274.  ASYMPTOTIC EXPANSIONS IN SEQUENTIAL ESTIMATION FOR  THE  FIRST
        ORDER RANDOM COEFFICIENT  AUTOREGRESSIVE  MODEL:  REGENERATIVE
        APPROACH

Vsevolod K. Malinovskii

     Edgeworth expansions  for the  distribution of  statistics
which  is  sequential  least  squares  estimator of $\beta =\bE
B_1$, $$U_{\nu(t)}=t^{-1}\sum_{i=1}^{\nu(t)} \frac{X_{i-1} X_i}
{1+\s^2  X_{i-1}^2}$$, where  $$\nu(t)=\min\Big \{n\geq 1:\sum_
{i=1}^n \frac{X_{i-1}^2}{1+\s^2X_{i-1}^2}\geq t\Big\}$$, in the
framework   of  a  first  order  ergodic   random   coefficient
autoregression (RCA) $X_n=B_n X_{n-1}+\epsilon_n, \qquad n=1,2,
\dots$,  with  $B_n$  and  $\epsilon_n$  i.i.d.  and   mutually
independent, are obtained in explicit form.
     The proof is based on regenerative  approach and  consists
in  two  steps.  First,  the  RCA  process  is   placed  in  an
appropriate  Markov chain context and the  splitting  technique
is used to extend  the underlying Markov chain to  regenerative
sequence. Consequently,  the asymptotical expansions  evaluated
for regenerative stopped  random sequences are  applicable.  On
the second step, going back  to the RCA process, the expansions
written  in  terms  of  regeneration  cycles,  are expressed in
terms of the initial RCA.

[email protected]

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275. AN ASYMPTOTICALLY 4-STABLE PROCESS

Krzysztof Burdzy and Andrzej Madrecki

The paper presents a construction of a process whose
state space is the space of infinite sequences of complex
numbers. The process is 4-stable in an appropriate
(asymptotic) sense. Its fourth variation is a linear
function of time. Its quadratic variation is (in a weak sense)
a Brownian motion.

[email protected] (plain TeX file available)

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276.  RADIAL DISTANCE OF BROWNIAN MOTI0N ON RIEMANNIAN MANIFOLD

M.Liao and W.A.Zheng


      Let $R_t$ be the radial distance of a Brownian motion in
an n-dimensional Riemannian manifold M starting at x and let
T be the first hitting time to the distance a. We show that
E[R_{t\wedge T}]^2=nt-(1/6)S(x)t^2+o(t^2), where S(x) is the scalar
curvature. The same formula holds for
E[R^2_t] under some boundedness condition on M.

[email protected] or [email protected] (Latex file available)

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277. A CASCADE DECOMPOSITION THEORY WITH APPLICATIONS TO MARKOV
     AND EXCHANGEABLE CASCADES

Edward C. Waymire and Stanley C. Williams

A multiplicative random cascade refers to a positive T-martingale in
the sense of Kahane on the ultrametric space $T = \{0,1,,\dots, b-1\}.$
A new approach to the study of multiplicative cascades is introduced.
The methods apply broadly to the problems of: (i) nondegeneracy
criterion, (ii) dimension spectra of carrying sets, and (iii) divergence
of moments criterion.  Specific applications are given to cascades
generated by Markov and exchangeable processes, as well as to 
homogeneous independent cascades.

[email protected]

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278. A LARGE DEVIATION RESULT WITHOUT CONTINUITY HYPOTHESES
     ON THE DIFFUSION COEFFICIENTS

Weian Zheng

Most literatures in large deviation theory of diffusion processes 
mainly treated the case where the diffusion coefficient is Lipschitz. However,
there are a lot of cases in applications we meet the diffusion processes
with non-continuous coefficients. 
       In this paper we reconsider S.R.S.Varadahan's large deviation result
$$\lim_{t\to 0}t\log H(x,t,y)=-{dis^2(x,y)\over 2}.            (*)$$
in the case where the infinitessimal operator is given by
$$Lf=\sum_{ij} g^{ij}(x){\partial^2\over \partial x_i\partial x_j}f
+\sum_{i}b_i(x){\partial \over \partial x_i}f$$
under the condition that at least one of the following two hypothesses
on $g^{ij}$ is true:
A)  there is a sequence of smooth matrix-valued functions
$g^{ij}_n(x)$ such that $$ g^{ij}_n(x)\searrow g^{ij}(x) .$$
B)  Suppose there is a sequence of smooth matrix-valued functions
$g^{ij}_n(x)$ such that $$ g^{ij}_n(x)\nearrow g^{ij}(x). $$
We show that the upper large deviation bound holds 
when Condition A) is true and  that the lower large deviation 
bound holds when Condition B) is true. As a conclusion, (*) holds when both 
A) and B) are satisfied.

[email protected] (LaTeX file available)

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279.  STOCHASTIC STABILIZATION AND DESTABILIZATION

Xuerong Mao

It has been observed that noise can be used to destabilize a 
given stable system but it can also be used to stabilize a 
given unstable system or to make a given stable system even
more stable. The pioneering work on stochastic stabilization
was done by Has'minskii in 1969. A more detailed study of 
stochastic stabilization of a linear system dx/dt = Ax via
Brownian motions (using Stratonovich integrtal) has been
carried out by many authors e.g. Arnold, Crauel, Wihstutz.
On the other hand, there is almost no work on the stochastic
stabilization of a non-linear system dx/dt = f(x, t). In this
paper we shall use stochastic perturbation described by linear
Ito integration w.r.t. Brownian motions to stabilize or 
destabilize the given non-linear system. It is shown in
this paper that any non-linear system dx/dt = f(x, t) can
be stabilized by Brownian motion provided |f(x, t)| <= K|x|
for some K>0. On the other hand, this system can also be
destabilzed if the dimension of the state is more than 1.
It should be stressed that in this paper we use Ito integral
rather than Stratonovich's.

[email protected]

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280. SIMULTANEOUS UNIQUENESS OF INFINITE CLUSTERS IN COUPLED PERCOLATION
     MODELS
 
Kenneth S. Alexander
 
 In processes such as invasion percolation and certain models of
continuum percolation, in which a possibly random label f(b) is attached to
each bond b of a possibly random graph, percolation models for various values
of the order parameter r are naturally coupled: one can define a bond b to be
occupied at level r if f(b) <= r.  If the labeled graph is stationary, then
under the mild additional assumption of positive finite energy, a result of
Gandolfi, Keane, and Newman (Prob. Th. Rel. Fields 92, 511 (1992)) ensures
that, in lattice models, for each fixed r at which percolation occurs, the
infinite cluster is unique a.s.  Analogous results exist for other particular
models.  A unifying framework is given for such fixed-r results, and it is
shown that if the site density is finite and the labeled graph has positive
finite energy, then with probability one, uniqueness holds simultaneously for
all values of r.  An example is given to show that when the site density is
infinite, positive finite energy does not ensure uniqueness, even for fixed
r. 

[email protected]

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281. ON HEAT CONDUCTION IN A COMPOSITE MEDIUM - A PROBABILITY
     APPROACH

Weian Zheng

  There are extensive studies on heat conduction in a 
composite medium. In one dimensional case, there is a general method  
by solving the boundary problem when medium is composed
of several pieces of diffrent materials. However, in the general
case, there is no unified method. For many years, mathematician and
phycian know that those problems are related to the second differential
operator in divergence form. Nevertheless, the precised discussion in terms 
of Dirichlet form was just recent. In this paper,we use the notion of 
diffusion process on a non-smooth Riemannian manifold to discuss the related 
boundary problem and limiting problem.

[email protected] (LaTeX file available)

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282. SHORTEST COMMON SUPERSTRINGS OF RANDOM STRINGS

Kenneth S. Alexander

     Given a finite collection of strings of letters from a fixed alphabet,
it is of interest, in the contexts of data compression and DNA sequencing, to
find the length of the shortest string which contains each of the given
strings as a consecutive substring.  In order to analyze the average behavior
of the optimal superstring length, substrings of specified lengths are
considered with the letters selected independently at random.  An asymptotic
expression is obtained for the savings from compression, that is, the
difference between the uncompressed (concatenated) length and the optimal
superstring length.

[email protected]

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283. A LARGE DEVIATION RESULT FOR A CLASS OF DIRICHLET PROCESSES

W.Zheng

Let C>0 be a constant and let 1/C <q(x)< C be a bounded
measurable function.
We consider the large deviation problem for the diffusion process
with infinitessimal operator
$${1\over q}{\partial\over \partial x_i}q{\partial\over \partial x_j}$$
We prove that its density function satisfies
$$\lim_{t\to 0}t\log H(x,t,y)=-{\vert x-y\vert^2\over 2}.$$

[email protected] (LaTeX file available)

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284. TRANSITION DENSITY ESTIMATES FOR BROWNIAN MOTION ON AFFINE NESTED 
     FRACTALS

P.J.Fitzsimmons, B.M.Hambly and T.Kumagai

A class of affine nested fractals is introduced which have different
scale factors for different similitudes but still have the symmetry 
assumptions of nested fractals. For these fractals estimates on the 
transition density for the Brownian motion
are obtained using the associated Dirichlet form. An upper bound for
the diagonal can be found using a Nash-type inequality, then probabilisitc
techniques are used to obtain the off-diagonal bound. The approach differs 
from previous treatments as it uses only the Dirichlet form and no
estimates on the resolvent. The bounds obtained are expressed in terms of
the intrinsic resistance metric on the fractal.

[email protected] (LaTeX file available)

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285.  SOME MARKOV PROPERTIES OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS.

Serge Cohen

In previous articles we were interested in SDE's on manifolds driven by non 
continuous semimartingales; we got an extension of Meyer-Schwartz's second 
order calculus. Here we  will be concerned with Markov properties  of 
solutions, when the driving process is a Levy process living in a vector 
space, more precisely we will compute their infinitesimal generator. As an 
application we are studying a process called pseudo alpha stable process 
constructed as  the stochastic development on a  Riemannian manifold 
of a vector valued alpha-stable process living in the tangent space of the 
manifold. We compare this pseudo alpha stable process with the alpha-stable 
process on the manifold  obtained as a Brownian motion time-changed by
a suitable subordinator. We  give a probabilistic proof that pseudo 
alpha-stable and  alpha-stable processes do not have the same laws on  suitable
Riemannian manifolds with a pole and a rotationally invariant metric. On the 
sphere in dimension 2, it is worth mentioning that   the 
infinitesimal generator of Brownian motion and of the pseudo 1-stable process 
are linked via a formula involving a concave, piecewise affine function. 
As a consequence of this remark the pseudo 1-stable process on the 2-sphere 
is not a Brownian motion time-changed by a subordinator.

[email protected] (Latex file available)

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286. ON THE POSITIVE HARRIS RECURRENCE FOR MULTICLASS QUEUEING
     NETWORKS: A UNIFIED APPROACH VIA FLUID LIMIT MODELS

J. G. Dai 

It is now known that under the usual traffic condition (the nominal
load being less than one at each station) is _not_ sufficient for
stability for a multiclass open queueing network.  Although there has
been some progress in establishing the stability for a multiclass
network, there is no unified approach to this problem.  In this paper,
we prove that a queueing network is positive Harris recurrent if the
corresponding fluid limit model eventually reaches zero and stays
there regardless the initial system configuration. As an application
to the result, we prove that single class network, multiclass
feedforward network and first-buffer-first-served preemptive resume
discipline in a reentrant line are positive Harris recurrent under the
usual traffic condition.

[email protected]

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287. CHUNG'S LAW OF THE ITERATED LOGARITHM FOR ITERATED BROWNIAN MOTION

Davar Khoshenvisan and Thomas M. Lewis

Let X and Y be independent standard Brownian motions.
For any non-negative t, we define the iterated Brownian motion,
Z, by setting Z(t) = X( |Y(t)| ). Recent work of K. Burdzy 
derive a law of the iterated logarithm for Z. In this paper,
we derive the other LIL (also known as Chung's LIL) for this
process.

[email protected]

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288. A UNIFORM MODULUS RESULT FOR ITERATED BROWNIAN MOTION

Davar Khoshnevisan and Thomas M. Lewis

Let X and Y be independent Brownian motions on the
real line with X(0)=Y(0)=0. Define iterated Brownian motion
as follows: Z(t)=X( Y(t) ). According to Funaki (1979),
the process, Z, relates to the square of the Laplacian
much like X does to the ordinary Laplacian. Burdzy (1993a,b)
contains a number of interesting facts about Z. In particular,
the LIL behavior of Z for t near zero. In this paper, we derive the 
corresponding exact uniform modulus of continuity of Z. 

[email protected]

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289.  DIVERGENT SUMS OVER EXCURSIONS
  
K. Bruce Erickson
        
Criteria for the almost sure divergence or convergence of sums
of functions of excursions away from a recurrent point in the 
state space of a Markov process are proved.  These are applied
to the excursions from 0 of reflecting diffusions, in particular,
reflecting Brownian motion, to derive some interesting sample
path properties for these processes.

[email protected]

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290. THE ASYMMETRIC CONTACT PROCESS ON A FINITE SET

Rinaldo Schinazi

We consider the asymmetric contact process on a finite set (with N sites)
on Z. The process on Z may have two distinct critical values. When this is the
case we show that between the two critical values (including the first
critical value and excluding the second one) the extinction time of the process
on the finite set is of order cN when N is large, where c is an explicit
constant. At the second critical value we show that the extinction time
is of order N^2. Below the first critical value and above the second critical
value, known results of Durrett and Liu and Durrett and Schonmann show that
the extinction time is of order log N and exp N, respectively.

[email protected]   (plain) Tex file available

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