- A Remark on Kac's Scattering Length Formula
- On General Perturbations of Symmetric Markov Processes
- Potential Theory of Moderate Makov Dual Processes
- Converse Jensen Inequality
- Martingale Functions of Brownian Motion and its Local Time
- Lévy Systems and Time Changes
- On Joint Fourier-Laplace Transforms
- Time Change Approach to Generalized Excursion Measures, and its Application to Limit Theorems
- Stochastic Calculus for Symmetric Markov Processes (Originally titled "Stochastic Calculus for Dirichlet Processes")
- Perturbation of Symmetric Markov Processes
- The Dirichlet Form of a Gradient-type Drift Transformation of a Symmetric Diffusion
- Drift Transformations of Symmetric Diffusions, and Duality
- On the Existence of Recurrent Extensions of Self-similar Markov Processes
- Excursion Theory Revisited
- A New Approach to the Martingale Representation Theorem
- Superposition Operators on Dirichlet Spaces
- Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process
- Absolute Continuity of Symmetric Markov Processes
- Non-symmetric Perturbations of Symmetric Dirichlet Forms
- Excursions Above the Minimum for Diffusions
- On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin
- Class of smooth functions in Dirichlet spaces

- I present a proof of A. Jakubowski's result that in the Chebyshev correlation inequality (cov[f(X)g(X)] >= 0
for increasing f and g) equality implies that either f(X) is a.s. constant or g(X) is a.s. constant.
Available as a pdf
(83k) file. [April 26, 2024]
- 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)*, and^{2}- t*Y(t)*are all local martingales, then^{3}-3t Y(t) -t*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*d*P*-*f*(*m*); If*Q*is a second probability measure such that d*Q*/d*P*<=*k*(a constant), then*V*(*Q*) <=*k**V*(*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*S*be the maximum to time_{t}*t*of |*B*|. For positive*p*define*Y*[_{t}=S^{p-2}_{t}*B*]^{2}_{t}-t*+cS*. We give a simple proof of the following result of B. Davis and J. Suh [On Burkholder's supermartingales,^{p}_{t}*Illinois J. Math.***50**(2006) 313--322]: Define*c*. Then (i) for_{0}= (2-p)/p*p*in (0,2],*Y*is a submartingale iff*c >= c*, and (ii) for_{0}*p*in [0,infinity),*Y*is a supermartingale iff*c < c*. Available in dvi (4K) and pdf (32K) formats. [September 1, 2006]_{0} - Let
*X*=(*X*)_{t}be a right Markov process with infinitesimal generator_{t >= 0}*L*and lifetime*z.*Let*A*= int_{t}_{0}^{t}*a(X*be a continuous additive functional of_{s}) ds*X*with*A*finite, and let_{z}*f(x)*be the expected value of exp(-*A*) as a function of the starting point_{z}*x=X*. We give a simple proof, based on the domination principle, that_{0}*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)*. Available in dvi (4K), postscript (29K), and pdf (193K) formats. [June 11, 2002]^{2}]/4 - We provide a short proof of the fact that if
*X*is a symmetric diffusion and*u(X*is a Dirichlet process (basically, a process of finite quadratic variation) then_{t})*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