Preprints
Short Notes
- A proof based on excursions below the maximum is given of J. Lehoczky's theorem on the the joint distribution of the drawdown time and the maximun up to that time, for a one-dimensional diffusion.
Available as a pdf
(187k) file. [March 13, 2020]
- We provide an alternative proof of the fact that Kemeny's constant, for a positive recurrent discrete-time Markov chain with countably infinite state space, is infinite.
Available as a pdf
(123k) file. [February 13, 2020]
- A simple proof of the observation that the tail-sum formula from probability theory holds for arbitrary measures.
Available as a pdf
(124k) file. [November 6, 2019]
- A simple proof of a general form of a result of A. Adamou and O. Peters
on the advantage of cooperation in certain multiplicative growth models.
Available as a
pdf (98K) file. [July 24, 2019]
- I provide a simple non-combinatorial proof of two integral identities of N. Kimura and O.G. Ruehr,
the subject of a recent note in the
American Mathematical Monthly by J.P. Allouche.
Available as a
pdf (106K) file. [June 26, 2019]
- A recent improvement on the discrete classic L^2 Hardy inequality, due to Keller, Pinchover, and Pogorzelski, is shown to follow
from a general Hardy inequality for Dirichlet forms.
Available as a
pdf (107K) file. [April 10, 2018]
- A filtering formula of G. Nappo and B. Torti, for the conditional distribution of reflected
Brownian motion at a fixed time given the history of its local time (at 0) up to that time,
is shown to be a special case of a general result in the excursion theory of Markov processes.
Available as a
pdf (48K) file. [June 28, 2012]
- I give a new proof of J. Wesolowski's characterization of the Poisson process: If X(t) is an
increasing cadlag process
with X(0)=0, such that Y(t) := X(t) - t, Y(t)2 - t,
and Y(t)3 -3t Y(t) -t
are all local martingales, then X(t) is a unit-rate Poisson process.
Available as a
pdf (40K) file. [June 28, 2012]
- I give a simple proof of the Dragomir-Jensen inequality: For a probability measure P with
mean m,
and a convex function f, define V(P) = int f dP - f(m);
If Q is a second probability measure such that dQ/dP <= k (a constant),
then V(Q) <= kV(P).
Available as a
pdf (40K) file. [March 29, 2012]
- I show how a method of D.W. Stroock can be adapted to show that if f is an additive mapping from
a Banach space E to another Banach space, and if f is measurable with respect to the completion of the Borel
sigma-field of E relative to any centered Gaussian meausre on E, then f is continuous and hence linear.
Available as a
pdf (56K) file. [February 21, 2012]
- I discuss a multi-player Gambler's Ruin problem of S. M. Ross, using martingale ideas familiar
from the classical Gambler's Ruin problem.
Available in dvi (8K)
and
pdf (36K) formats. [January 6, 2010]
- Let B be a standard one-dimensional Brownian motion
and let St be the maximum to time t of |B|.
For positive p define Yt=Sp-2t
[B2t-t]+cSpt.
We give a simple proof of the following result of B. Davis and J. Suh
[On Burkholder's supermartingales, Illinois J. Math. 50 (2006) 313--322]:
Define c0 = (2-p)/p. Then
(i) for p in (0,2], Y is a submartingale iff c >= c0,
and (ii) for p in [0,infinity), Y is a supermartingale iff c < c0.
Available in dvi (4K)
and
pdf (32K) formats. [September 1, 2006]
- Let X=(Xt)t >= 0
be a right Markov process with infinitesimal generator
L and lifetime z.
Let At = int0t a(Xs) ds
be a continuous additive functional of X with Az
finite, and let f(x) be the expected value of exp(-Az) as a function
of the starting point x=X0. We give a simple proof,
based on the domination principle, that
f is the minimal solution with values in [0,1] of the equation Lf=af.
This extends a recent result of
P.K. Pollett and V.T. Stefanov [Path integrals for continuous-time Markov
chains. J. Appl. Prob. 39 (2002) 901-904.]
Available in dvi (5K)
and
pdf (192K) formats. [June 13, 2003]
- Using Skorokhod stopping, we prove the converse of Jensen's inequality:
If X and Y are integrable random variables with
E[f(X)] greater than or equal to
E[f(Y)]
for every convex function f, then there are random variables
X' and Y' equal in distribution to X and X respectively such that
Y'=E[X'|Y']. Available in dvi (4K), postscript
(29K), and
pdf (179K) formats. [June 11, 2002]
- We give a simple proof of Grüss's inequality:
If X is a bounded random variable with lower bound m and upper bound M,
then the variance of X is bounded above by [(M-m)2]/4. Available in dvi (4K), postscript
(29K), and
pdf (193K) formats. [June 11, 2002]
- We provide a short proof of the fact that if X is a symmetric
diffusion and u(Xt) is a Dirichlet process (basically, a process of
finite quadratic variation) then u is locally in the Dirichlet
space of X. Available in dvi (12K), postscript
(76K), and
pdf (76K) formats. [July 19, 2001]
- We present short proof that (aside from a deterministic
non-atomic component) a completely random measure in the sense of Kingman
[Completely random measures, Pacific J. Math., 21 (1967) 59-78.
MR:
35#1079] is purely atomic, almost surely. Available in dvi (5K), postscript
(49K), and
pdf (160K) formats. [December 15, 2000]
- We present a simple approach to the approximation of
a Markov process by pure jump processes, complementing recent work of
Z.-M. Ma, M. Röckner, and T.-S. Zhang
[Approximation of arbitrary Dirichlet processes by Markov chains.
Ann. Inst.
H. Poincaré Probab. Statist. 34 (1998) 1--22. MR:
99c:60160]
and
Z.-M. Ma, M. Röckner, and W. Sun [Approximation of Hunt process by
multivariate Poisson processes. Preprint, 1999]. Available in dvi (10K), postscript
(72K), and
pdf (86K) formats. [April 20, 2000]
- Using an integral representation theorem
of the author [Skorokhod embedding by randomized hitting times, In:
Seminar on Stochastic Processes 1990, Birkhäuser,
Boston, pp. 183-191], we give a short proof of a result of
Mokobodzki, characterizing the extreme points of the set of
stopping distributions of a right Markov process. Available in dvi (4K), postscript
(40K), and
pdf (59K) formats. [November 18, 1999]
- We give a probabilistic proof of a recent
result of A. Dunkels [A note on the equilibrium potential of
certain Dirichlet spaces. Potential Anal. 6 (1997)
99--104], giving an explicit identification of the equilibrium
measure of a Borel set on its (fine) interior. Dunkels dealt with
symmetric Lévy processes in Euclidean space; our argument
works for general symmetric Markov processes. Available in dvi (7K), postscript
(54K), and
pdf (102K) formats. [November 2, 1999]
- In a recent paper [Orthogonal measures and
absorbing sets for Markov chains, Math. Proc. Camb. Phil.
Soc. 121 (1997) 101--113], P.-D. Chen and R.L. Tweedie
make use of the natural embedding of a separable and separated
measurable space in the compact cube {0,1}N. The point of
this note is to show that a well-known representation of functions
measurable with respect to the sigma-algebra generated by a
sequence of functions can be substituted for the rather intricate
arguments they employ. Available in dvi (8K), postscript
(56K), and
pdf (82K) formats. [March 24, 1999]
- If the sum of independent real-valued
random variables is normally distributed, then so are the summands
(Cramer-Lévy theorem). In this note an analogous result is
proved for Brownian motion: If the sum of independent local
martingales is Brownian motion, then the summands are (modulo
deterministic time changes) also Brownian motion. Available in pdf
form (116K).
[September 29, 1995; corrected March 28, 2019]
- A new (?) proof of the fact that the
Lebesgue measurable solutions of the functional equation f(x+y) =
f(x) + f(y) are of the form f(x) = cx. In postscript form
(45K) and in
pdf form (3K). [September 29, 1995]
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March 13, 2020