Let $\{X_i\}$ be a sequence of independent, identically distributed nondegenerate random variables and $S_n = \sum^n_{i = 1}X_i$. We consider the question for various centering sequences $\{\alpha_n\}$: when is it possible to find a positive, monotone sequence $\{\beta_n\}$ such that $\lim \sup \beta^{-1}_n (S_n - \alpha_n) = c$ a.s., $c$ a finite nonzero constant? If $\alpha_n = \operatorname{med} S_n$, we obtain a necessary and sufficient condition for this. An important corollary is a one-sided version of the Hartman-Wintner law of the iterated logarithm: if $E(X^+)^2 < \infty$, then it is always possible to find such a norming sequence. Explicit norming sequences are given which are easy to obtain. Necessary and sufficient conditions are also given for being able to find a norming sequence $\{\beta_n\}$ for the two-sided problem $(\lim \sup \beta^{-1}_n |S_n - \alpha_n| = c$ a.s.) when $\alpha_n = ES_n$ and $\alpha_n = 0$. The two-sided problem with $\alpha_n = \operatorname{med} S_n$ was solved by Kesten. The one-sided problem remains open for $\alpha_n = ES_n$ and $\alpha_n = 0$. Examples are given which illustrate the advantage of considering different centering sequences. A one-sided version of Strassen's converse to the law of the iterated logarithm is also given: if $\lim \sup S_n/ \sqrt{2n \log \log n} = 1$ a.s., then $EX = 0, EX^2 = 1$.

If $X$ is a bounded left-continuous and piecewise constant process and if $Z$ is an arbitrary process, both adapted, then the stochastic integral $\int X dZ$ is defined as usual so as to conform with the sure case. In order to obtain a reasonable theory one needs to put a restriction on the integrator $Z$. A very modest one suffices; to wit, that $\int X_n dZ$ converge to zero in measure when the $X_n$ converge uniformly or decrease pointwise to zero. Daniell's method then furnishes a stochastic integration theory that yields the usual results, including Ito's formula, local time, martingale inequalities, and solutions to stochastic differential equations. Although a reasonable stochastic integrator $Z$ turns out to be a semimartingale, many of the arguments need no splitting and so save labor. The methods used yield algorithms for the pathwise computation of a large class of stochastic integrals and of solutions to stochastic differential equations.

Let $A_t$ be a nonadapted continuous additive functional of a right continuous strong Markov process $X_t$, and let $\tau_t$ denote the right continuous inverse of $A_t$. We give general sufficient conditions for the time-changed process $X_{\tau_t}$ to again be a strong Markov process with a new transition semigroup. We give several examples and show that birthing a process at a last exit time and killing a process at a cooptional time may be realized as raw time changes.

A stochastic solution of the Neumann problem is obtained, when the second order elliptic operator $L$ is degenerate at the boundary of the domain. Let $D$ be a domain in $R^n$ with the smooth boundary $\partial D$, and the second order elliptic operator $L$ be defined in $R^n$. We construct a diffusion $X^r(t) = (\mathscr P\underset{\smile}^r, D(\mathscr P\underset{\smile}^r))$ in $\bar{D}$ such that (i) $D(\mathscr P\underset{\smile}^r) \supset D(A^r) = \{f \in C^2(D); \partial f/\partial v = 0$ for $x \in D\}$, (ii) $f \in D(A^r) \Rightarrow \mathscr P\underset{\smile}^r f = Lf$. With that diffusion, the stochastic solution of our Neumann problem is defined, and the existence and the uniqueness conditions of that are obtained. The analytic meaning of our stochastic solution is explained. The diffusions in $\bar{D}$, satisfying the other Venttsel boundary conditions are also constructed, which are useful for the degenerate third boundary value problems.

Let $X_1, X_2, \cdots$ be independent identically distributed (i.i.d.) random variables with a continuous distribution function. Let $R_0 = 0, R_k = \min \{j: j > R_{k - 1}$ and $X_j > X_{j + 1}\}$ and $T_k = R_k - R_{k - 1}, k \geq 1$. We prove that a process $T^{(n)} = \{T_{k + n}\}^\infty_{k = 1}$ converges, in the sense of distribution functions, exponentially fast to a strongly mixing ergodic process. It is shown that $(\max_{1 \leq k \leq n} T_k)/\log n(\log \log n)^{-1} \rightarrow 1$ almost surely and in $L_p, p > 0$. Also, the number of runs $T_k, 1 \leq k \leq n$, larger than or equal to some $m$ is proven to be Poisson distributed in the limit, if $n/m$! converges to a positive number.

The convergence properties of the empirical characteristic process $Y_n(t) = n^{1/2}(c_n(t) - c(t))$ are investigated. The finite-dimensional distributions of $Y_n$ converge to those of a complex Gaussian process $Y$. First the continuity properties of $Y$ are discussed. A class of counterexamples is presented, showing that if the underlying distribution has low logarithmic moments then $Y$ is almost surely discontinuous, and hence $Y_n$ cannot converge weakly. When the underlying distribution has high enough moments then $Y_n$ is strongly approximated by suitable sequences of Gaussian processes with specified rate-functions. The approximation is based on that of Komlos, Major and Tusnady for the empirical process. Convergence speeds for the distribution of functionals of $Y_n$ are derived. A Strassen-type log log law is established for $Y_n$, and supremum-functionals on the appropriate set of limit points are explicitly computed. The technique throughout uses results from the theory of the sample function behaviour of Gaussian processes.

A class of first-order autoregressive processes is given for which the extreme value limit theorems of Loynes and Leadbetter do not apply. A limit theorem is derived for these processes that depends on the parameter $r$, an integer greater than or equal to 2.

Let $X(n)$ be the largest observation in a random sample of size $n$ from a standard normal population. In this paper we investigate the limiting behavior of the distribution and the moments of $X(n)$ for large $n$. The results of this paper provide rates of convergence of the distribution and the moments of $X(n)$.

This paper uses randomization to prove the existence of a subset of the circle which is small in the sense that its Lebesque measure is small, and which is large in the sense that a certain stochastic process almost surely visits the set infinitely often.

Analogues of the Marcinkiewicz-Zygmund and Rosenthal inequalities for Banach space valued random vectors are proved. As an application some results on the strong law of large numbers are obtained. It is proved that the Marcinkiewicz SLLN holds for every $p$-integrable, mean zero $B$-valued $\mathrm{rv}$ if and only if $B$ is of Rademacher type $p(1 \leq p < 2)$.

Two types of multidimensional processes are defined. The first of these generalizes a univariate IFRA process due to Ross and relates to a multivariate concept of IFRA due to Esary and Marshall. The second of these relates to a multivariate concept of IFRA due to the present authors.

A secretary problem is an optimal stopping problem based on relative ranks. To the usual formulation of the secretary problem we add a cumulative interview cost function $h(\cdot)$, no longer obtaining "cutoff point" rules. For an appealing form of $h(\cdot)$ we examine the limiting results using the infinite secretary problem. It is shown that the other appealing form of $h(\cdot)$ leads to trivial limiting results. A large class of problems is considered and recursive equations leading to the limiting solution are given. In particular we solve the problem of minimizing expected rank with a linear interview cost function. An approximation to the rank problem with fixed cost $c$ per interview is obtained (for all values of $c$) through the solution of a single differential equation.