This is a guide to some recent work in the theory of martingale inequalities. Methods are simplified; some new proofs are given. A number of new results are also included.

This is a survey on sample function properties of Gaussian processes with the main emphasis on boundedness and continuity, including Holder conditions locally and globally. Many other sample function properties are briefly treated. The main new results continue the program of reducing general Gaussian processes to "the" standard isonormal linear process $L$ on a Hilbert space $H$, then applying metric entropy methods. In this paper Holder conditions, optimal up to multiplicative constants, are found for wide classes of Gaussian processes. If $H$ is $L^2$ of Lebesgue measure on $R^k, L$ is called "white noise." It is proved that we can write $L = P(D)\lbrack x\rbrack$ in the distribution sense where $x$ has continuous sample functions if $P(D)$ is an elliptic operator of degree $> k/2$. Also $L$ has continuous sample functions when restricted to indicator functions of sets whose boundaries are more than $k - 1$ times differentiable in a suitable sense. Another new result is that for the Levy(-Baxter) theorem $\int^1_0(dx_t)^2 = 1$ on Brownian motion, almost sure convergence holds for any sequence of partitions of mesh $o(1/\log n)$. If partitions into measurable sets other than intervals are allowed, the above is best possible: $\mathscr{O}(1/\log n)$ is insufficient.

Given two equivalent Gaussian processes the notion of a non-anticipative representation of one of the processes with respect to the other is defined. The main theorem establishes the existence of such a representation under very general conditions. The result is applied to derive such representations explicitly in two important cases where one of the processes is (i) a Wiener process, and (ii) a $N$-ple Gaussian Markov process. Radon-Nikodym derivatives are also discussed.

Three sufficient conditions for continuity of real-valued, separable, Gaussian processes on $\mathbb{R}^1$ are compared. They are: (1) Fernique's (1964) integral condition, (2) the Kahane (1960)-Nisio (1969) condition on the spectrum of stationary processes and (3) Dudley's (1967) condition involving $\varepsilon$-entropy. Let $S_1 \equiv$ set of stationary, separable, Gaussian processes that can be proven continuous by condition $i = 1, 2, 3$. Dudley (1967) has shown that $S_1 \subseteq S_3$. It is shown here that $S_2 \subset S_1 \subset S_3$, that is, the inclusion is strict. These results extend to non-stationary processes where appropriate. The Kahane-Nisio condition is strengthened and the best possible integral condition for continuity involving the spectrum is given. A result on the $\varepsilon$-entropy of blocks in a separable Hilbert space is also of independent interest.

An occupation-time density is identified for a class of absolutely continuous functions $x(t)$ in terms of $x'(t)$ and the number of times that $x(t)$ assumes the values in its range. This result is applied to stationary random processes with a finite second spectral moment. As a by-product, a generalization of Rice's formula for the mean number of crossings is obtained.

Let $W^{(N)}$ denote the $N$-parameter Wiener process, that is a real-valued Gaussian process with zero means and covariance $\Pi^N_{i=1} (s_i \wedge t_i)$ where $s = \langle s_i\rangle, t = \langle t_i\rangle, s_i \geqq 0, t_i \geqq 0, i = 1, 2, \cdots N$. Then $W^{(N,d)}$ is to be the process with values in $R^d$ determined by making each component an $N$-parameter Wiener process, the components being independent. Our concern is with continuity and recurrence properties of the sample functions. In particular we give integral tests for upper functions which reduce in the case $N = d = 1$ to the integral tests of Kolmogorov, and of Chung-Erdos-Sirao. We formulate and prove precise statements of the fact that $W^{(N,d)}$ is interval recurrent (point recurrent) if and only if $d \leqq 2N (d < 2N)$.

Let $K_r$ be the set of $r$-tuples $\mathbf{k} = (k_1, k_2, \cdots, k_r)$ with positive integers for coordinates $(r \geqq 1)$. Let $\{X_k: \mathbf{k} \in \mathbf{K}_r\}$ be a set of i.i.d. random variables with mean zero, and let $\leqq$ denote the coordinate-wise partial ordering on $K_r$. Set $|\mathbf{k}| = k_k k_2 \cdots k_r$ and define, for $\mathbf{k} \in K_r: S_k = \sum_{j\leqq\mathbf{k}} X_j$. If $\{E_k : \mathbf{k} \in K_r\}$ is a set of events indexed by $K_r$, we say (given $\omega$) "$E_k$ f.o." if $\mathbf{\exists} \mathbf{I}(\omega) \in K_r$ such that $\mathbf{k} \nleqq \mathbf{I}$ implies $\omega \in E_k^c$. We say "$E_k$ a.l." if given any $\mathbf{I} \in K_r, \exists \mathbf{k} \geqq \mathbf{I}$ such that $\omega \in E_k$. We prove: (i) If $E\{|X_k|(\log^+ |X_k|)^{r-1}\} = \infty$, then given any $A > 0, P\{|S_k|/|\mathbf{k}| > A \text{a.1.}\} = 1$. Using martingale techniques, we also give a new proof of the converse result due to Zymund: (ii) If $E\{|X_k| (\log^+ |X_k|)^{r-1}\} < \infty$, then given any $\varepsilon > 0, P\{|S_k|/|\mathbf{k}| < \varepsilon \text{f.o.}\} = 1$. For non-identically distributed independent random variables with mean zero, the usual conditions sufficient for convergence of $S_n/n$ to zero in the linearly ordered case are also sufficient for matrix arrays.

A new proof is given for the inequality $$E(\Phi(\Sigma^\infty_{\nu=1} E(z_\nu \mid \mathscr{F}_\nu))) \leqq CE(\Phi(\Sigma^\infty_{\nu=1} z_\nu)),$$ where $z_1, z_2, \cdots, z_n, \cdots$ are nonnegative random variables on a probability space $(\Omega, \mathscr{F}, \mathbf{P}), \mathscr{F}_1 \subset \mathscr{F}_2 \subset \cdots \subset \mathscr{F}_n \subset \cdots \mathscr{F}$ is a sequence of $\sigma$-fields and $\Phi(u)$ is a convex function satisfying $\Phi(2u) \leqq c\Phi(u)$.

A minimax procedure is found for filtering the "signal" from the "noise" in a stationary time series when it is known only that the spectral distribution function of the "signal" lies in a convex set defined by linear inequalities.

A tightness condition for families of Gaussian processes with continuous paths is given and applied to the question of convergence of sums of independent Gaussian processes.

Under a slight regularity condition we prove that if the spent and residual waiting times at $t$ in a renewal process are independent random variables for one value of $t = t_0$, then the process is Poisson.

This note extends a theorem of Welsch (1971) on the joint asymptotic distribution of some order statistics of a strong-mixing, stationary, Gaussian sequence.

Given $N$ points distributed at random on $\lbrack 0, 1)$, let $p_n$ be the size of the smallest interval that contains $n$ points. Previous work finds $\operatorname{Pr}(p_n \leqq p)$, for $n > N/2$, and for $n \leqq N/2, p = 1/L, L$ an integer. This paper finds the distribution of $p_n$, for all $n, N,$ and $p$.