The contention that artificial neural nets operate by a process of distributed hypothesis testing is supported by analysis of the antiphon, a model which is close to standard associative-memory models but is intended primarily to represent memory storage under stochastic disturbance of the system. The memory capacity of the antiphon under "neuronal" inference rules has been evaluated elsewhere; here it is evaluated under the supposition of efficient inference procedures.

We consider a modification of the spatial epidemic with removal that has regrowth of susceptibles. We show that if the original epidemic is supercritical, then the modified process has a nontrivial stationary distribution.

Inspection and repair policies for $(n - r + 1)$-out-of-$n$ systems are compared stochastically with respect to two partial orderings $\geq^{b_1}$ and $\geq^{b_2}$ on the set of permutations of $\{1, 2, \ldots, n\}$. The partial ordering $\geq^{b_1}$ is finer than the partial ordering $\geq^{b_2}$. A given permutation $\pi$ of $\{1,2, \ldots, n\}$ determines the order in which components are visited and inspected. We assume that the reliability of the $i$th independent component is given by $P_i$, where $P_1 \leq P_2 \leq \cdots \leq P_n$. If $\pi$ and $\pi'$ are two permutations such that $\pi \geq^{b_1} \pi'$, then we show that the number of inspections necessary to achieve minimal or complete repair is stochastically smaller with $\pi$ than with $\pi'$. We also consider three policies for minimal repair when the components are each made up of $t$ "parts" assembled in parallel. It is shown that if $\pi \geq^{b_2} \pi'$, then the number of repairs necessary under $\pi$ is stochastically smaller than the number necessary under $\pi'$, but that in general this is not true for the finer ordering $\geq^{b_1}$. The results enable one to make interesting comparisons between various inspection and repair policies, as well as to understand better the relationship between the orderings $\geq^{b_1}$ and $\geq^{b_2}$ on the set of permutations.

Aldous and Shepp recently proved that the Erlang distribution of a given order is the least variable phase-type distribution of that order, in the sense of minimizing the coefficient of variation. Here we prove that it is also least variable in the sense of majorization. We give an example showing that the result does not extend in the obvious way to general distributions with rational transforms and this suggests that the inequality hinges on the Markov property.

Consider some model of random finite trees of increasing size. It often happens that the subtree at a uniform random vertex converges in distribution to a limit random tree. We introduce some structure theory for such asymptotic fringe distributions and illustrate with many examples.

The theory of random utility maximization for a finite set of alternatives is generalized to alternatives which are elements of a compact metric space $T$. We model the random utility of these alternatives ranging over a continuum as a random process $\{Y_t, t \in T\}$ with upper semicontinuous (usc) sample paths. The alternatives which achieve the maximum utility levels constitute a random closed, compact set $M$. We specialize to a model where the random utility is a max-stable process with a.s. usc paths. Further path properties of these processes are derived and explicit formulas are calculated for the hitting and containment functionals of $M$. The hitting functional corresponds to the choice probabilities.

For matrices with iid columns, the expectations of the title are examined using ideas from convex geometry. In particular, a representation is shown which involves random realizations from the class of convex bodies called zonoids. This result is used to derive bounds and comparisons of various types.

Let $X_1, X_2, \ldots$ be a sequence of independently and identically distributed integer-valued random variables. Let $Y_{t - m + 1,t}$ for $t = m, m + 1,\ldots$ denote a moving sum of $m$ consecutive $X_i$'s. Let $N_{m,T} = \max_{m \leq t \leq T} \{Y_{t - m + 1,t}\}$ and let $\tau_{k,m}$ be the waiting time until the moving sum of $X_i$'s in a scanning window of $m$ trials is as large as $k$. We derive tight bounds for the equivalent probabilities $P(\tau_{k,m} > T) = P(N_{m,T} < k)$. We apply the bounds for two problems in molecular biology: the distribution of the length of the longest almost-matching subsequence in aligned amino acid sequences and the distribution of the largest net charge within any $m$ consecutive positions in a charged alphabet string.