Comparative probability (CP) is a theory of probability in which uncertainty is measured by a CP ordering of events, rather than by a probability measure. A CP order is additive iff it has an agreeing probability measure. This paper deals with the formation of joint CP orders from given marginals, both with and without a certain independence condition, and with emphasis on the nonadditive case. Among the results are these: a CP model for many independent and identically distributed trials of a single experiment must be additive, with an agreeing probability measure of product type; there are CP marginals that have no joint CP order at all; there is a class of CP models, strictly containing all the additive ones, which are well behaved with respect to the formation of joint orders. We present as well several sufficient conditions, and one necessary condition, under which given marginals have a joint CP order.

Structure of semimarkov processes in the sense of Cinlar (1975) will be clarified by relating them to Chung processes. Start with a semimarkov process. For each attractive instantaneous state whose occupation time is zero, dilate its constancy set so that the occupation time becomes positive; this is achieved by a random time change. Then, mark each sojourn interval of an unstable holding state $i$ by $(i, k)$ if its length is between $1/k$ and $1/(k - 1)$; this is "splitting" the unstable state $i$ to infinitely many stable states $(i, k)$. Finally, replace each sojourn interval (of the original stable states $i$ plus the new stable states $(i, k)$) by an interval of exponentially distributed length. The result is a Chung process modulo some standardization and modification.

Let $a = \{f(\cdot, \theta): \theta \in J\}, J$ an interval, be a family of univariate probability densities (with respect to Lebesgue measure) on an interval I. First, a necessary and sufficient condition is proved for $\mathcal{a}$ to be identifiable whenever $\mathcal{a} \subset C_0(J)$, the class of continuous functions on $J$ vanishing at $\infty$. If $f_G$ is a $G$-mixture of the densities in $a$ with $G$ unknown, an estimator $G_n$ based on $f_G$ and $\mathscr{B} = \{f(x, \bullet): x \in I\}$ is provided such that $G_n \rightarrow_w G$ under certain conditions on $a$. If $X_1, \cdots, X_n$ are i.i.d. random variables from $f_G$, an estimator $\hat{G}_n$ is provided such that $G_n(X_1, \cdots, X_n, \cdot) \rightarrow_w G(\bullet)$ almost surely under certain conditions on $a$ and $G$. Furthermore, it is shown that $|f_{G_n}(x) - f_G(x)| \rightarrow 0$ a.s. and in $L_2$ with rates like $O(n^{-C}) (C > 0)$ under certain conditions on the density estimator $\hat{f}_G(x)$ involved in the definition of $\hat{G}_n$. The conditions of various theorems are verified in the case of location parameter and scale parameter families of densities.

Let $X_1, X_2, \cdots$ be independent random variables such that $EX_n = 0, EX_n^2 = 1, n = 1,2, \cdots$ and the uniform Lindeberg condition is satisfied. Let $S_n = X_1 + \cdots + X_n$. In this paper, we study the first exit time $N_c = \inf \{n \geqq m: |S_n| \geqq cb(n)\}$ for general lower-class boundaries $b(n)$. Our results extend the theorems of Breiman, Brown, Chow, Robbins and Teicher, Gundy and Siegmund who studied the case $b(n) = n^{\frac{1}{2}}$. We also obtain the limiting moments of $N_c$ in the case $b(n) = n^\alpha (0 < \alpha < \frac{1}{2})$ as analogues of recent results in extended renewal theory.

Probabilities of moderate deviations of the sample mean from the population mean are calculated for certain strictly stationary $\phi$-mixing processes.

Let $W^{(N, d)}$ be the $N$-parameter Wiener process with values in $R^d$. It is shown that almost all sample functions of $W^{(N, d)}$ have dimensional number $2N$ and zero $2N$-measure when $d \geqq 2N$. Our results extend earlier ones of Taylor for $N = 1$.

A theory of stochastic integral equations is developed for the integrals of Kunita, Watanabe, and P. A. Meyer. Existence and uniqueness of solutions of systems of equations with semimartingale (or "quasi-martingale") differentials is proved, in which we include as particular cases the customary results as put forth by McKean and Gihman and Skorohod. Under weaker conditions we prove existence and uniqueness with explosions, and study the explosion times. We show that when the (random) coefficients or the differentials converge, the solutions converge to the solution of the limiting equation.

Consider a recurrent random walk $\{X_n\}$ with state space $S \subseteqq R^d (d \leqq 2)$. A stopping time $T$ is called subterminal if it satisfies a technical condition which essentially states that it is the first time a path possesses some property which does not depend on how long the process has been running. Suppose $T$ is a subterminal time which can occur only when $\{X_n\}$ is in a bounded set; then under an additional assumption a ratio limit theorem (as $n \rightarrow \infty$) is obtained for $P(T > n \mid X_0 = x) (x\in S)$. The theorem applies in particular to the hitting time of a bounded set with nonempty interior in the general case, and to the hitting time of a bounded set with nonzero Haar measure in the nonsingular case.

The interrelationships among conditions for convergence in law of sequences of likelihood ratios and the concept of contiguity are explored. Related results of Le Cam (1960), Hajek and Sidak (1967) and Roussas (1972) are extended, modified and clarified. In particular, it is shown that if likelihood ratios converge in law under the numerator hypothesis, then they converge under the denominator hypothesis and the hypotheses are contiguous (numerator to denominator).

We show how a recent theorem of J. Hoffmann-Jorgensen and G. Pisier can be formulated in such a way as to include a theorem of N. Jain and M. Marcus. We also obtain some central limit theorems on $L^p\lbrack 0, 1\rbrack$, for $1 \leqq p < 2$.

Almost sure convergence of generalized $U$-statistics and von Mises' differentiable statistical functions is studied with the help of the general $L \log L$ martingale convergence theorem.

We show that there is a one-to-one correspondence between invariant measures for the noncritical multitype Galton-Watson process and invariant measures for the single type process with a linear offspring probability generating function. Two corollaries emerge as simple applications, the first being Spitzer's Martin boundary representation, the second giving the asymptotic behaviour of the measures. Both require no extra moment assumptions and are valid for the multitype theory.

We give an elementary proof of the decomposition of a subadditive stochastic process as an additive process plus a positive subadditive process with time constant 0. The proof is based on two ideas. The first is a general idea for obtaining a kind of weak limit point for $L_1$-bounded sequences of random variables, based on the martingale convergence theorem. The second is a general result about martingales which seems to be new and is of independent interest.

A martingale inequality for the $\rho_q$ distance from the uniform empirical process to zero is proved, compared with other inequalities for the process, and used to establish a law of the iterated logarithm.

Let $(X, \mathscr{B}(X), P)$ be a probability space and let $T$ be a $K$-automorphism. If $T$ satisfies a Rosenblatt mixing condition of a certain kind, we show that if $\{k_n\}^\infty_{n=1}$ is an arbitrary increasing sequence of integers and $g$ belongs to a certain class of functions then $$\lim_{n\rightarrow\infty} \frac{1}{n} \sum^n_{j=1} g(T^{k_j}x) = E(g) \mathrm{a.s.}$$