Probability Abstracts 2

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12. A NECESSARY CONDITION FOR MAKING MONEY FROM FAIR GAMES

H. Kesten and G. Lawler

60J

Let $X_1,X_2,\ldots$ be independent random variables such that $X_j$
has distribution $F_{\sigma(j)}$ where $\sigma(j) = 1$ or $2$,
and the distributions $F_i$ have man zero.  Assume that $F_i$ has
a finite $q_i^{th}$ moment for some $1 < q_i < 2$.  Let $S_n =
\sum_{j=1}^n X_j$.  We show that if $q_1 + q_2 > 3$, then
$\limsup P\{S_n > 0\} > 0$ and \lim sup P\{S_n < 0 \} < 0$ for each
sequence of ones and twos.

jose@math.duke.edu

13. ESCAPE PROBABILITIES FOR SLOWLY RECURRENT SETS

G. Lawler

60J

A set $A \subset Z^d (d \geq 3)$ is defined to be slowly recurrent for simple
random walk if it is recurrent but the probability of entering 
$A \cap \{z: n < |x| \leq 2n \}$ tends to zero as $n \rightarrow \infty$.
A method is given to estimate escape probabilities for such sets, i.e., the
probability of leaving the ball of radius $n$ without entering the set.
The methods are applied to two examples.  First, half-lines and finite
unions of linew in $Z^3$ are considered. The second example is a random
walk path in four dimensins. In the latter case it is proved that the
probability that two random walk paths reach the ball of radius $n$
without intersecting is asymptotic to $c(\ln n)^{-1/2}$, improving a
result of the author.


jose@math.duke.edu

14. L-SHAPES FOR THE LOGARITHMIC $\eta$-MODEL FOR DLA IN THREE DIMENSIONS

G. Lawler

60K

The logarithmic $\eta$-model for DLA is the nearest neighbor cluster model
in which points are added to the cluster with a probability proportional
to harmonic measure raised to the $C \ln n$ power.  For $C$ sufficiently
large, it is shown that in $Z^3$ such a cluster can form an $L-shape$
with positive probability.  In this case, the ratio of the legs of the
cluster depends on the value of $C$.


jose@math.duke.edu

15. ASYMPTOTICS FOR EUCLIDEAN MINIMAL SPANNING TREES ON RANDOM POINTS

David Aldous and J. Michael Steele

60K

Asymptotic results for the Euclidean minimal spanning tree 
on n random vertices in R^d can be obtained from consideration of
a limiting infinite forest whose vertices form a Poisson process
in all R^d.
In particular we prove a conjecture of Robert Bland:
the sum of the d'th powers of the edge-lengths of the 
minimal spanning tree
of a random sample of n points from the uniform distribution in the
unit cube of R^d tends to a constant as n tends to infinity.

Whether the limit forest is in fact a single tree is a hard open
problem, relating to continuum percolation.

aldous@stat.berkeley.edu

16. EXTREMAL CHARACTER OF THE LYAPUNOV EXPONENT OF THE STOCHASTIC HARMONIC OSCILLATOR

Mark A. Pinsky, Northwestern University

We give a formula for the quadratic Lyapunov exponent of the harmonic
oscillator in the presence of a finite-state Markov process. In case the 
noise process is reversible, the quadratic Lyapunov exponent is stricly 
less than for the corresponding white noise process obtained 
through the central limit theorem . An example is presented of a 
non-reversible Markov noise process for which  this inequality is reversed.

m_pinsky@math.nwu.edu

17. RIESZ DECOMPOSITIONS AND SUBTRACTIVITY FOR EXCESSIVE MEASURES

P. J. Fitzsimmons and R. K. Getoor

60J45

The convex cone of excessive measures of a right Markov process is an
example of a superharmonic semigroup in the abstract potential theory developed
by Arsove and Leutwiler. We show that their theory of Riesz decompositions can
be sharpened in the case of excessive measures. In particular there is always a
Riesz decomposition relative to a given potential cone (resp. harmonic cone).
An element of an ordered convex cone is {\it subtractive} if each majorant is a
specific majorant. This notion of subtractivity features prominently in the
theory of harmonic cones. We give a complete characterization of the
subtractive elements in the cone of excessive measures.

pfitzsim@ucsd.bitnet (plain.TeX file available)

18. ON CHOQUET'S DICHOTOMY OF CAPACITY FOR MARKOV PROCESSES

 P. J. Fitzsimmons and Mamoru Kanda

 60J45

Following Choquet,  the capacity associated with a Markov process
is said to be dichotomous if each compact set K contains two disjoint sets with
the same capacity as K. In the context of right processes, we prove 
that the dichotomy of capacity is equivalent to Hunt's hypothesis that
semipolar sets are polar. We also show that a weaker form of the
dichotomy is valid for any L\'evy process with absolutely continuous
potential kernel.

pfitzsim@ucsd.bitnet (plain.TeX file available)

19. DIFFUSION APPROXIMATION FOR AN AGE-STRUCTURED POPULATION

A. Bose and I.Kaj

60J

We prove a diffusion limit theorem in the sense of weak convergence
of measure-valued processes for a simple population age model first
studied by Kendall. We show that in the diffusion limit scaling, the
population structured in age groups behaves in the same way as the
total population size, but with an exponential weight. A particular
feature fo the limiting process is that in general it is discontinuous
at time zero.
 
amit_bose@carleton.ca
 

20. MEASURE-VALUED AGE-STRUCTURE PROCESSES

A. Bose and I. Kaj

60G, 60J

We study the age-structure in an age-dependent branching population as
a measure-valued Markov jump process. The process is characterized by
its Laplace transform function which is the mild solution of a non-
linear differential equation, sometimes referred to as an "S-equation".
In particular, we study this process under the assumption that the branching
mechanism is critical. In this case we consider a diffusion approximation
scaling and determine explicitly the weak limit in terms of a continuous
state branching process.
 
amit_bose@carleton.ca
 

21. ON A FIRST PASSAGE PROBLEM FOR BRANCHING BROWNIAN MOTIONS

I. Kaj and P. Salminen

60J

Let $\omega=(\omega_t,t)_{t\ge0}$ be a space-time realization, i.e.
a tree, of a critical or subcritical one-dimensional branching Brownian
motion. For a tree $\omega$ and $x\ge0$ denote with $Z_z(\omega)$ the
number of particles, which are located for the first time on the vertical
line through $(x,0)$ and, which do not have an ancestor on this line.
In this note we study the process $Z=\{Z_x:\, x\ge0\}$. It is seen that
$Z$ is a Galton-Watson process, and its creation rate and offspring
distribution are computed. Here we use ideas of Neveu, who considered
a similar problem in the supercritical case. Moreover, in the critical
case the continuous state branching processes obtained as weak limits
of the processes $Z$ are characterized.
 
ingemar_kaj@carleton.ca    until June 15, 1991
mat_ik@tdb.uu.se           thereafter
 

22. SIZE-BIASED SAMPLING OF POISSON POINT PROCESSES AND EXCURSIONS

Mihael Perman, Jim Pitman and Marc Yor.

60J

Some general formulae are obtained for size-biased sampling from a 
Poisson point process in an abstract space where the size of a point
is defined by an arbitrary strictly positive function.
These formulae explain why in certain cases (gamma and stable) the size-biased
permutation of the normalized jumps of a subordinator can be represented by a
stickbreaking (residual allocation) scheme defined by independent beta random
variables.  An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on
the Brownian and Bessel cases when the associated inverse local time is a 
stable subordinator.  Results in this case are linked to generalizations of
the arcsine law for the fraction of time spent positive by Brownian motion.

pitman@stat.berkeley.edu

23. AN EMBEDDING OF COMPENSATED COMPOUND POISSON PROCESSES WITH APPLICATIONS TO LOCAL TIME

Davar Khoshnevisan   

60J,60F

We present a Brownian embedding for a broad class of 
compensated compound Poisson processes. Applications 
of this method are discussed for a problem of level
crossings, as well as Donsker's invariance-type of
principles. In particular, we give a central limit 
theorem for local times, via strong approximations.

davar@math.washington.edu 

24. LEVEL CROSSINGS OF THE EMPIRICAL PROCESS

Davar Khoshnevisan    

60J, 62G

This is an in-depth study of asymptotics for the 
number of times a linear empirical process crosses 
the true distribution function are studied in depth. 
We identify the the limiting as the local time of 
Brownian bridge, and give a process version of this. 

davar@math.washington.edu 

25. MOMENT INEQUALITIES FOR FUNCTIONALS OF BROWNIAN CONVEX HULL

Davar Khoshnevisan    

60J, 60E

We give an extension of the Burkholder-Davis-Gundy 
inequality for a large class of monotone functionals 
of the convex hull of the path of a d-dimensional
Brownian motion, up to a stopping time.

davar@math.washington.edu  (LaTeX file available )

26. WHEN DOES THE RAMER FORMULA LOOK LIKE THE GIRSANOV FORMULA?

M. Zakai and O. Zeitouni

60G30,60H

Let (B,H,P) be an abstract Wiener space and for every real r let
T(r,w)=w+rF(w)  be a transformation from B to B. It is
well known that under certain assumptions the measures induced
by T(r,w) and its inverse  are mutually absolutely continuous with respect
to P and the density function is represented by the Ramer formula.  In this
formula, the Carleman-Fredholm determinant associated with r grad F 
appears as a factor.  We characterize the class of  F for which 
the Carleman Fredholm determinant equals 1, namely the 
case where the Ramer expression reduces to the familiar Girsanov
form. The proof is based on a characterization of quasi-nilpotent
Hilbert-Schmidt operators.

zeitouni@ee.technion.ac.il

27. LIMIT DISTRIBUTIONS OF U-STATISTICS RESAMPLED BY SYMMETRIC STABLE LAWS

Jerzy Szulga

60E 62H 60F 60H 62G 10C

We prove that the limit distributions of  U-statistics
$$
n^{-1/\alpha}\SUM_{1\le i_1,\ldots,i_d\le n}
Z_{i_1}\cdots Z_{i_d}\, f(V_{i_1},\ldots,V_{i_d}),
$$
where $(Z_i)$ are independent copies of a random variable belonging to the 
domain of normal attraction of a symmetric $\alpha$-stable law, $0<\alpha<2$, 
coincide with probability laws of multiple stable integrals
$$
X^df=\mint f(t_1,\ldots,t_d)\,dX(t_1)\ldots dX(t_d).
$$

szulga@auducvax.bitnet/szulga@ducvax.auburn.edu (LATEX file available)

28. SOME LIMIT THEOREMS FOR RESAMPLED U-STATISTICS INVOLVING MULTIPLE L\'EVY INTEGRALS

Jerzy Szulga

60E 62H 60F 60H 62G 10C

We investigate weak limits of Poisson multiple inegrals and apply the results 
to show that weak limits of certain U-statistics are multiple Poisson 
integrals or, more generally, multiple L\'evy integrals.

szulga@auducvax.bitnet/szulga@ducvax.auburn.edu (LATEX file available)

29. THE ISOMORPHISM BETWEEN MULTIPLE STABLE INTEGRALS

Jerzy Szulga

60H, 60E, 46B

It is shown that a study of the existence and some geometric 
properties of multiple stable integrals for various indices 
$\alpha,\, 0<\alpha<2$, can be reduced to the corresponding properties of a 
multiple stable integral for a fixed, arbitrarily chosen, index of stability.

szulga@auducvax.bitnet/szulga@ducvax.auburn.edu (LATEX file available)

30. SERIES EXPANSIONS OF MULTIPLE L\'EVY INTEGRALS

Jerzy Szulga

60H 60E 46B

Multiple stochastic integrals with respect to an infinitely
divisible symmetric random measure without a Gaussian component admit
LePage-type representations by means of certain multiple random series.

szulga@auducvax.bitnet/szulga@ducvax.auburn.edu (LATEX file available)

31. CALCUL STOCHASTIQUE AVEC SAUTS SUR UNE VARIETE

J. Picard

60G

Consider a continuous semimartingale X(t) on a manifold; if we are given a
connection on the manifold, then we can construct the Ito integral along
X(t) of a predictable process in the cotangent bundle; in particular we
can say that X(t) is a martingale if all Ito integrals along X(t) are real
local martingales. In this paper, we propose an extension of these
definitions to the case of semimartingales with jumps; in particular we
can handle discrete-time processes. We also propose a definition of a
stochastic transport along the semimartingale.

(To appear in Seminaire de Probabilites XXV)

picard@ucfma.univ-bpclermont.fr

32. A REMARK ON THE PROOF OF ITO'S FORMULA FOR $C^2$ FUNCTIONS OF CONTINUOUS SEMIMARTINGALES

W.S. Kendall

60H

The It\^o formula is the fundamental theorem of stochastic calculus. This
short note presents a new proof of It\^o's formula for the case of
continuous semimartingales. The new proof is more geometric than
previous approaches, and has the particular advantage of generalizing
immediately to the multivariate case without extra notational complexity.

w.s.kendall@cu.warwick.ac.uk

33. CONVEX GEOMETRY AND NONCONFLUENT $\Gamma$-MARTINGALES I: TIGHTNESS AND STRICT CONVEXITY

W.S. Kendall

60G

It is explained how four problems in convexity, $\Gamma$-martingales,
harmonic maps, and geodesic theory are all related. A conjecture by
Emery amounts to asking whether in fact all four problems are
equivalent! A limited equivalence is established between two of the
problems, using a tightness theorem for $\Gamma$-martingales to show
the following: if the $\Gamma$-martingale Dirichlet problem for a
compact domain {\cal B} has unique solutions and if {\cal B} supports a
strictly convex function then there is a convex function
${\cal Q} : {\cal B}\times{\cal B} \to [0,1]$
vanishing only on the diagonal. In short, {\cal B} then hs {\it convex
geometry}.

I now have a counterexample for the Emery conjecture -- preprint
available. W

w.s.kendall@cu.warwick.ac.uk

34. CONVEX GEOMETRY AND NONCONFLUENT $\Gamma$-MARTINGALES II: WELL-POSEDNESS AND $\Gamma$-MARTINGALE CONVERGENCE

W.S. Kendall

60G

In the terminology of ``Convex Geometry and Nonconfluent
$\gamma$-martingales I'' a closed domain {\cal B} of a manifold
furnished with a connection $\Gamma$ is said to have \sl convex
geometry\rm, or property (${\bf A}$), if there is a bounded nonnegative
$\Gamma$-convex function {\cal Q} defined on ${\cal B}\times{\cal B}$
vanishing only on the diagonal.  It is said to have property (${\bf
B}$) if solutions to its Dirichlet problem for $\Gamma$-martingales are
unique and well--posed  (depend continuously on their limiting values
at time $\infty$) when they exist. In this paper it is shown that
(${\bf A}$) and (${\bf B}$) are equivalent if {\cal B} is compact,
strengthening Theorem 3.2 of Kendall (to appear). In the course of the
proof a result of independent interest is established: if  the limits
at time $\infty$ of nontrivial $\Gamma$-martingales in {\cal B} are
never nonrandom, and if the associated $\Gamma$-martingale Liouville
property is well-posed, then all $\Gamma$-martingales in {\cal B}
converge as time tends to infinity.

w.s.kendall@cu.warwick.ac.uk

35. RANDOM WALKS, ELECTRICAL RESISTANCE, AND NESTED FRACTALS

M. T. Barlow

One of the problems left open by Lindstrom in [Brownian motion on nested
fractals, Mem. AMS 420] was the uniqueness of the process. Using the
connection between symmetric Markov chains and electrical resistance, the
problem is reduced to one concerning the electrical resistance of networks.
(One has to consider resistance in a more general form than the usual 2-point
function). The problem is then solved for some (infinite) families of nested
fractals.

mtb3@phx.cam.ac.uk  (Plain Tex)

36. RANDOM WALKS AND DIFFUSIONS ON FRACTALS

M.T. Barlow

This is a survey of the field "diffusions on fractals", and concentrates
on the mathematicaL literature, with specific reference to heat-kernel
estimates.

mtb3@phx.cam.ac.uk   (Plain Tex file available)

37. ON THE EXISTENCE OF ORDERED COUPLINGS OF RANDOM SETS --- WITH APPLICATIONS

Tommy Norberg

60D05  (60B05, 60G99)

Let \( \psi \) and \( \varphi \) be two given random closed sets in a 
locally compact second countable topological space \( S \). (They need 
not be based on the same probability space.) The main result gives 
neccessary and sufficient conditions on the distributions of \( \psi \) 
and \( \varphi , \) for the existence of two random closed sets
\( \hat{\psi} \) and \( \hat{\varphi} , \) based on the same probability 
space and such that their distributions coincide with those of \( \psi \) 
and \( \varphi , \) resp., and \( \hat{\psi} \subseteq \hat{\varphi} \) a.s.

This coupling result tells us in particular when a probability distribution 
on \( S \) is selectionable w.r.t.\ (the distribution of) a random closed set.
An existence result for realizable thinnings of a simple point process is 
obtained by specializing it to supports of random measures.

The coupling result is extended to random variables in a countably based 
continuous poset, examples of which we mention various kinds of random 
capacities --- in particular random measures --- and random compact 
(saturated) sets. This extension moreover tells us when a probability 
distribution on \( S \) is selectionable w.r.t.\ (the distribution of) a 
random compact (saturated) set.

tommy@math.chalmers.se		(Latex file available)

38. UNIFORM QUADRATIC VARIATION FOR GAUSSIAN PROCESSES

Robert Adler and Ronald Pyke

60G     

We study the uniform convergence of the quadratic
variation of Gaussian processes, taken over large families of curves
in the parameter space.  A simple application of our main result shows  
that the quadratic variation of the Brownian sheet along all rays issuing 
from a point in  [0,1] converges uniformly (with probability one) as long 
as the meshes of the partitions defining the quadratic variation do not 
decrease too slowly. Another application shows that previous quadratic 
variation results for Gaussian processes on [0,1] actually hold uniformly 
over large classes of partitioning sets.

pyke@math.washington.edu

39. RANDOM SPLITTIGS OF AN INTERVAL

Tom Mountford and Sid Port
 
60K

Points are independently and uniformly distributed on the unit interval.
The first n-1 points subdivide the interval into n segments.For l a function
of n the events that the n-th point falls into the l-th smallest segment
occurs infinitely often with probability one or zero.A necessary and sufficient
condition is given for it to be one.When this is the case various limit results
for the number of occurences by time n are given,such as laws of large numbers,
clt,Poisson.The key fact is a uniform slln for the length of the l-th largest
segment.

scp@math.ucla.edu

40. SPITZER'S FORMULA INVOLVING CAPACITIES

Sid Port

60J

Asymptotic expansions of order 4 are given for the expected volume of a
stable sausage.

scp@math.ucla.edu

41. LOCAL TIMES AND RELATED SAMPLE PATH PROPERTIES OF CERTAIN SELF-SIMILAR PROCESSES

N.Kono and N-R,Shieh

60G

For certain self-similar processes with stationary increments, we 
prove the existence of temporally continuous local times and prove
the infinity of both the local oscillations of sample paths and the 
Hausdorff measures of level sets. We also prove that linear fractional
stable processes with stability parameters >= 1 are locally non-
deterministic and have jointly continuous local times. Moreover, we
prove that for certain self-similar processes with small scaling 
parameters the local times are differentiable in the space variable.

kono@platon.kula.kyoto-u.ac.jp

42. HOLDER CONTINUITY OF SAMPLE PATHS OF SOME SELF-SIMILAR STABLE PROCESSES

Norio Kono and Makoto Maejima

60G

The Holder continuity of sample paths of some classes of self-similar 
processes is examined. The results are (1) a general result for self-similar 
stable processes with stationary increments and (2) a sharper result for a 
special class of those processes which is that of harmonizable fractional 
stable motions.

maejima@math.keio.ac.jp

43. TRIMMED SUMS OF MIXING TRIANGULAR ARRAYS WITH STATIONARY ROWS

Makoto Maejima and Yuko Morita

60F

Limit laws of trimmed sums are studied for triangular arrays of row-wise
stationary random variables.  It is shown that if the marginal distribution
of the array belongs to the domain of an infinitely divisible law without
Gaussian component, the trimmed sum converges weakly to a nondegenerate
random variable under some mixing and local dependence conditions.

maejima@math.keio.ac.jp

44. A NOTE ON LINEAR AND HARMONIZABLE FRACTIONAL STABLE MOTIONS

Makoto Maejima and Narn-Rueih Shieh

60G

Cambanis and Maejima (1989) proved that the law of linear fractional stable
motion (which is self-similar alpha-stable) is distinct from that of real
harmonizable fractional stable motion (which is also self-similar alpha-
stable) when 1 < alpha < 2.  In this note, it is shown that the same 
statement remains true when 0 < alpha <,= 1.

maejima@math.keio.ac.jp (AMSTex file available)