This paper deals with numerous variants of bounds for probabilities of large deviations of sums of independent random variables in terms of ordinary and generalized moments of individual summands. A great deal of attention is devoted to the study of the precision of these bounds. In this connection comparisons are made with precise asymptotic results. At the end of the paper various applications of the bounds for probabilities of large deviations to the strong law of large numbers, the central limit theorem and to certain other problems are discussed.

Almost sure and in-probability representations of the empirical process by appropriate Gaussian processes are obtained when unknown parameters of the underlying distribution function are estimated. As to the method of estimation, we consider maximum likelihood and maximum likelihood-like estimators and construct the above-mentioned representations under a null hypothesis. Similar results are obtained also when using Durbin's more general class of estimators under a sequence of alternatives which converge to the null hypothesis. The resulting Gaussian processes depend, in general, on the true value of the unknown parameters.

In this paper, we study a nonstationary dynamic programming model $\{(S_n, \mathscr{S}_n), (A_n, \mathscr{A}_n), D_n, q_n, u: n \geqslant 1\}$ with standard Borel state spaces $(S_n, \mathscr{S}_n)$ and action spaces $(A_n, \mathscr{A}_n)$, upper semicontinuous admissible-action correspondences $D_n$, weakly continuous transition laws $q_n$, and Borel measurable total reward function $u: S_1 \times A_1 \times \cdots \rightarrow R_-$. We establish existence of a persistently optimal (degenerate) plan for this model under regularity and boundedness assumptions on conditional expectations of $u$, but require no special separable form of $u$ such as intertemporal additivity. The methods of proof utilize results on weak convergence of probability measures and selection theorems in the context of optimization of functions over correspondences. We give two characterizations of the persistent optimality of a feasible plan: the first that the plan be both thrifty and equalizing, and the second that the plan satisfy an optimality criterion that entails period-by-period optimality.

This paper studies belief functions, set functions which are normalized and monotone of order $\infty$. The concepts of continuity and condensability are defined for belief functions, and it is shown how to extend continuous or condensable belief functions from an algebra of subsets to the corresponding power set. The main tool used in this extension is the theorem that every belief function can be represented by an allocation of probability--i.e., by a $\cap$ -homomorphism into a positive and completely additive probability algebra. This representation can be deduced either from an integral representation due to Choquet or from more elementary work by Revuz and Honeycutt.

Algorithms for linear interpolator and interpolation error for a minimal univariate weakly stationary stochastic process with discrete multiparameter are derived. The Fourier coefficients of the inverse of the spectral density play an important role in the determination of these algorithms.

We give necessary and sufficient conditions for tail $\sigma$-fields of time-homogeneous one-dimensional diffusion processes and birth and death processes to be trivial and study related questions.

If $g$ is the transform of a martingale $f$ under a predictable sequence $v$ uniformly bounded in absolute value by 1, then $$\lambda P(g^\ast \geqslant \lambda) \leqslant 2\|f\|_1, \lambda > 0$$, and this inequality is sharp.

Let $X$ be the Brownian motion process in $\mathbb{R}^d, d \geqslant 3$ with $X(0) = 0$. Let $L_r$ be the last exit time of $X$ from the ball of radius $r$ centered at the origin. Then $(L_r)$ has independent increments and we compute the distribution of $L_r$. When $d = 3$ this yields a simple proof of a recent result of Pitman.

Consider a sequence of independent and identically distributed random variables attached to the points of an independent point process $P$. The random record process is the epochs of successive maxima in this sequence. In this paper necessary and sufficient conditions are found to ensure a certain tail behaviour for the distributions of times to successive records, and interrecord times, when $P$ is a renewal process with relatively stable partial sums for its intervals.

The duality-related point of view of Hoeffding for i.i.d. large deviations is extended to the Markov case. This reconciles Koopmans' "function-analytic" view of Markovian large deviations with Boza's "information theoretic" view, and extends the validity of the latter to the not-necessarily-finite case.

Let $\mathscr{P}$ be the family of all stationary information sources with alphabet $A$. Let $F: \mathscr{P} \rightarrow(-\infty, \infty)$ be convex and upper semicontinuous in the weak topology. It is shown that for $n = 1,2, \cdots$, there is an estimator $Y_n: A^n \rightarrow (-\infty, \infty)$, such that if $\mu \in \mathscr{P}$ is ergodic and the process $(X_1, X_2,\cdots)$ has distribution $\mu$, then $Y_n(X_1,\cdots, X_n)\rightarrow F(\mu)$ in $L^1$ mean.

Let $(\Omega, \mathscr{a}, P)$ be a probability space and $\mathscr{B} \subset \mathscr{a}$ be a $\sigma$-field. Let $s$ with $1 < s < \infty$ be fixed. If $f \in L_s(\Omega, \mathscr{a}, P)$ and $\mathscr{B} \subset \mathscr{a}$ is a $\sigma$-field, the unique element $g_f \in L_s(\Omega, \mathscr{a}, P)$ such that $\|f - g_f\|_s = \inf\{\|f - h\|_s: h \in L_s(\Omega, \mathscr{B}, P)\}$ is called the $s$-predictor of $f$ relative to the $s$-norm and the $\sigma$-field $\mathscr{B}$. Such $g_f$ exists and is uniquely determined. The mapping $P_s^\mathscr{B}: f\rightarrow g_f$ is called a prediction operator. The prediction operator is not necessarily a linear operator. The problem is to characterize the $\sigma$-fields $\mathscr{B}$ in terms of $P\mid\mathscr{a}$ for which $P_s^\mathscr{B}$ is a linear operator. We show that, for a fixed $\sigma$-field $\mathscr{B}$, the prediction operators $^\mathscr{B}_s$ are linear for all $s$ or for no $s \neq 2$. We give a necessary and sufficient condition for the linearity of $s$-predictors in terms of conditional expectations only. If moreover regular conditional probabilities given $\mathscr{B}$ exist, the $s$-predictors are linear if and only if the regular conditional probabilities of $P\mid\mathscr{a}$ given $\mathscr{B}$ consist only of measures concentrated on at most two points. Furthermore we obtain a simple criterion that $s$-prediction coincides with the usual conditional expectation (i.e., with 2-prediction): the conditional expectations of indicator functions may only assume the values $0, \frac{1}{2}$ and 1.

Analogues are proposed for the concepts of self-decomposability and stability for distributions on the nonnegative integers. It turns out that these "discrete self-decomposable" and "discrete stable" distributions have properties that are quite similar to those of their continuous counterparts.

The probability of covering [0, 1) at least $m$ times by $N$ randomly placed subintervals of length $p$ is derived for all $m, N, p$.

Let $X$ be an $L_p$ martingale, $3 \leqslant p < \infty$. Let $M = \sup|X_k|$ and $V^2 = \Sigma(X_k - X_{k-1})^2$. We show that $\|X\|_p \leqslant (p - 1)\|V\|_p$ and, consequently, that $\|M\|_p \leqslant p\|V\|_p$.