## ABSTRACT

This extraordinary compilation is an expansion of the recent American Mathematical Society Special Session celebrating M. M. Rao's distinguished career and includes most of the presented papers as well as ancillary contributions from session invitees. This book shows the effectiveness of abstract analysis for solving fundamental problems of stochas

### chapter |4 pages

#### An Appreciation of my teacher, M.M. Rao

ByJ. A. Goldstein

ByM. L. Green

ByJerry Uhl

ByM. M. Rao

### chapter |3 pages

#### M. Rao in this space. As recalled above, a semimartingale F(x, t) is repre

F. If {X, F, t F is a continuous semimartingale, x D, whose local characteristics > 0, then the integral in (53) can be defined with Itô integral is
Byis a in IR with continuous paths, then under the conditions for some = [0 = t < t < ... < t = T]

### chapter 2|4 pages

#### Applications of Sinkhorn balancing to counting problems

ByIsabel Beichl, Francis Sullivan

### chapter 3|4 pages

#### Zakai equation of nonlinear filtering with Ornstein-Uhlenbeck noise: Existence and Uniqueness

ByAbhay Bhatt, Belram Rajput, Jie Xiong

ByM. Burgin

### chapter 6|3 pages

#### Invariant Sets for Nonlinear Operators

ByGiséle Ruiz Goldstein, Jerome A. Goldstein

### chapter 8|1 pages

#### Approximating Scale Mixtures

ByHassan Hamdan, John Nolan

g/ |) |
By|g(x|

By1|| c

### chapter c|3 pages

#### f. [32, p. 98]) the notion of a regular stationary numerical

K –1, is regular stationary if the component se- are regular and jointly stationary in the sense that the means and n, collections C}, .
ByII.2 A numerical vector sequence z, with K components z for any Z} and {k,k,... ,k

### chapter |8 pages

#### the results of Bass [2] may be applied because the pre-Hilbert M (x) is a subspace of the Marcinkiewicz space M of sequences in the previous section. To see this, suppose zÎM(x), then

. The theorem of Bass but also that if y and z H (x) N – 1, then µ(z) = (z, 1), m = N – 1 exist for every z
Byof the type (3.19). But if the limit for z certainly the limsup exists and so z if y Î H (x) as a limit. | is a continuous invertible linear map in the same manner as Proposition II.4 for and so we omit the proof. (x) = (x, 1) exists form = 0, 1,. Î H(x). is simple and thus omitted, (see for the continuous time case and [5] for a review), gives the connection and cyclostationary sequences. It is the founda- for various representations of stochastic cyclostationary sequences and

### chapter 12|15 pages

#### Connections Between Birth-Death Processes

ByAlan Krinik, Carrie Mortensen, Gerardo Rubino

P(t) =
By+...+A

### chapter |3 pages

#### fH(R), we have,

C\\Af\\ = A : X
Byis clear. • a second-order process {xt,t to a normed space X. Let Y be an n — an a to consider the process y = Ax by y(t,w) = , t T, w . First, we have to make sure

### chapter |2 pages

#### Proof. The process x induces a random element

ByE [ (j = E [ (, x

### chapter 14|2 pages

#### Moving Average Representation and Prediction for Multidimensional Harmonizable Processes

ByMarc H. Mehlman

### chapter |1 pages

#### if it has the following full rank virile covariance representation t) =

t, 1 j n} . .,Z B let j n,
By: s t, 1 j n} the closure is taken in t as the space of observables of X. be an ] for every {Z( ' is i IV.2 A random process, X, is deterministic if and only if

### chapter IV|2 pages

#### 5 A function c : D M is maximal if and only if c(.) H(D) and,

c(.) is a n – m full rank matrix valued function that is maximal, H(t) =
By(t) for all t R, and *(•')) = m(• •')I where m(d ) is Lebesgue measure on R, and only if X is purely nondeterministic.

### chapter 15|10 pages

#### Double-Level Averaging on a Stratified Space

ByNatella V. O'Bryant

### chapter C|2 pages

#### **,C > 0, such that

By(af)([X]) ||f||

### chapter |1 pages

#### the riskless asset or the market portfolio. Similarly, the allocation

pt = [p, ..., is the vector of excess returns p = z – z for
By+ z,t)W + x'p k = 0,1,2, ...,T – 1 = t, t, ..., t. Therefore, any risk averse investor will choose a strategy the solutions of problem (26) which maximizes his/her

### chapter |4 pages

#### In the most general case we can solve the above problem a defined joint distribution of the return vector. However, in the the vectors of risky returns z = [z,...,z]' t = 0, t,...,t are sta-

,... for any j are the deterministic vari- W – x' e where W
By(27) any fixed mean and initial wealth W. The multiperiod portfolio policies the risky assets xt = [x in the riskless return at time t

### chapter |1 pages

#### f is the density of Y. Therefore, when unlimited short selling is

= x'E(z)
Byis a fund separation model whose optimal are given by the optimization problem (19) or equivalently, by the we can solve the optimization

### chapter |2 pages

#### the following allocation problem c (H q))

E(W) - cP(W -VaR) =
Bytry to the

ByZ D. Ren

### chapter |1 pages

#### For instance, see [6], [59],[70], Prediction problems

X, the to X in the in H A}. is of the of the
Byby Niemi [45],[46],[48]. the prediction and filtering problems as follows. an asymptotically stationary process on {X : s a in the of the in

### chapter |1 pages

#### G is either of Type I, Type II, or Type III, as its group von Neumann algebra W*(G) = C*(G)** is of Type I,

G × G is also of Type I, so following Rao [61] we can modify the × such that T
Byin §3.1 as follows. Let G be of Type I. We call the continuous process {X(t), Î G} strongly harmonizable if K Î B(G), so X is strongly harmonizable a measurable mapping T on is a

Bya,f ,

### chapter 20|8 pages

#### Doubly Stochastic Operators and the History of Birkhoff s Problem 111

BySheila King, Ray Shiflett

### chapter |2 pages

#### of bounded real sequences is denoted by l , with norm || = sup |x|. Note that fÎl, with q < Þ |x| 0, and that l Ì

l. If f = |x| and ||f|| < . l l T
ByLet P= (p) be a given infinite stochastic matrix and let Tf = fP let Tf = fP = T is positive and linear. It follows that ||Tf|| = all terms are positive. So ||Tf|| ||T|| 1. Finally, from absolute convergence, STf = S (S

### chapter |6 pages

#### U 0 = see that f(W)

Bythe way W was expanded the countably infinite to the uncountable case. Until now, the study has to work with operators induced by matrices, and stochastic prop- in terms of finite or infinite row and column sums. In the

### chapter 21|3 pages

#### Classes of Harmonizable Isotropic Random Fields

ByRandall J. Swift

### chapter |4 pages

#### ' the representation of the covariance becomes ( t) = 2 J ( ) = r s – dF(

s – dF(
Byis the representation of a stationary isotropic covariance obtained by in spherical-polar form for the covariances was also given by Swift in

### chapter |12 pages

#### of individuals of each species per unit time. Since µ multiplies k, there is no loss of generality in assuming the ratio of µ and d to be as µ and d tend to 0.

Byd) – u(x, ) and v(x, y,t + d) – v(x, y,t)

### chapter 23|7 pages

#### Approximating the Time Delay in Coupled van der Pol Oscillators with Delay Coupling

ByStephen A. Wirkus