Motivated by several results and open problems concerning Harris' basic contact process, we consider the relationship between the critical behavior of the finite and infinite versions of Spitzer's reversible nearest particle systems. We show that the critical values for the finite and infinite systems agree, but that the behavior of the two systems at the common critical value can differ. The Nash-Williams recurrence criterion for reversible Markov chains is an important tool used in the proofs of the main results, and we give a new treatment of that theory. Finally, we compute several critical exponents for the nearest particle systems.

Consider a branching process in which particles are located in $\mathbb{R}^d$, do not move during their life times, die according to the exponential holding law, and at their deaths give birth to random number of particles which are located at distances from their parents. The total number process is supposed supercritical. We are interested in the number of particles living in a shifted region $D + tc$, denoted by $Z_t(D + tc)$, where $c \in \mathbb{R}^d$ and $D$ is a bounded set of $\mathbb{R}^d$, and observe a.s. convergences of $Z_t(D + tc)/E\lbrack Z_t(D + tc)\rbrack$ as $t \rightarrow \infty$. The result is applied to an associated non-linear evolution equation, which reduces, in a special case, to the equation of a deterministic model of simple epidemics.

A set of $n$ points in Euclidean space is partitioned into the $k$ groups that minimize the within groups sum of squares. Under the assumption that the $n$ points come from independent sampling on a fixed distribution, conditions are found to assure asymptotic normality of the vector of means of the $k$ groups. The method of proof makes novel application of a functional central limit theorem for empirical processes--a generalization of Donsker's theorem due to Dudley.

Balls are successively thrown, independently and uniformly, in $n$ given urns. Let $N_{n,m}$ be the number of throws required to obtain at least $m$ balls in each urn. Let $N'_{n,m,r}$ be the number of urns containing exactly $r$ balls upon completion of the $N_{n,m}$th throw, $r \geq m$. We prove that, given $N_{n,m} = \lbrack n \log n + (m - 1)n \log \log n + nx\rbrack, N'_{n,m,r} \sim e^{-x} (\log n)^{r-m+1}/r!$ as $n \rightarrow \infty$ in probability. From this, we derive the following limit law for the joint distribution of $N_{n,1}, \cdots, N_{n,m}: \lim_{n \rightarrow\infty} P(N_{n,i} \leq n \log n + (i - 1)n \log \log n + nx_i; 1 \leq i \leq m) = \prod^m_{i=1} \exp(-(1/(i - 1)!)e^{-x_i})$. This result generalizes earlier work of Erdos and Renyi who obtained the limit law for $N_{n,m}$ as $n \rightarrow \infty$.

For $X_1, X_2,\cdots$ i.i.d., $EX_1 = \mu \neq 0, S_n = X_1 + \cdots + X_n$, the asymptotic behavior of moments and error rates of the two-sided stopping rules $\inf \{n \geq 1: |S_n| > cn^\alpha\}, c > 0, 0 \leq \alpha < 1$, is considered. Convergence of (normalized) moments of all orders as $c \rightarrow \infty$ is obtained, without the higher moment assumptions needed in the one-sided case of extended renewal theory (Gut, 1974), and in a more general setting than just the i.i.d. case. Necessary and sufficient conditions are given for convergence of series involving the error rates, in terms of the moments of $X_1$.

Standard renewal theory is concerned with expectations related to sums of positive i.i.d. variables, $S_n = \sum^n_{i=1} Z_i$. We generalize this theory to the case where $\{S_i\}$ is a Markov chain on the real line with stationary transition probabilities satisfying a drift condition. The expectations we are concerned with satisfy generalized renewal equations, and in our main theorems, we show that these expectations are the unique solutions of the equations they satisfy.

Let $X_0, X_1 \cdots$ be a Markov process with transition function $p(x, dy)$. Let $L_n(\omega, \cdot)$ be its average occupation time measure, i.e., $L_n(\omega, A)= 1/n \cdot \sum^{n-1}_{i=0} \chi A(x_i(\omega))$. A powerful theorem concerning the lower bound of the asymptotic behavior of $L_n(\omega, \cdot)$ was proved by Donsker and Varadhan when $p(x, dy)$ satisfies a homogeneity condition. This paper tries to extend their results to some cases where such a homogeneity condition is not satisfied. This particularly includes symmetric random walks and Harris' chains.

In this note we prove weak invariance principles for some classes of mixing sequences of $L_2$-integrable random variables under the condition that the variance of the sum of $n$ random variables is asymptotic to $\sigma^2n$ where $\sigma^2 > 0$. One of the results is simultaneously an extension to nonstationary case of a theorem of Ibragimov and an improvement of the $\varphi$-mixing rate used by McLeish in his invariance principle for nonstationary $\varphi$-mixing sequences.

Berry-Esseen bounds of order $O(N^{-1/2})$ are established for simple linear rank statistics under the null-hypothesis. The theorems are proved for a wide class of scores generating functions which include the normal quantile function. This improves earlier results under the null-hypothesis in von Bahr (1976), Huskova (1977, 1979) and Ho and Chen (1978).

Let $X_{n1} < X_{n2} < \cdots < X_{nn}$ denote the order statistics of an $n$-sample from the distribution with density $f$. We prove the strong consistency and asymptotic normality of estimators based on the series $(\frac{1}{2}) \sum^{n-k}_1 (X_{n,r+k} + X_{nr})/(X_{n,r+k} - X_{nr})^p \text{and} \sum^{n-k}_1 (X_{n,r+k} - X_{nr})^{-p}$, where $k > 2p > 0$ are fixed constants. These series may be used to estimate functionals of $f$. The ratio of the series was introduced by Grenander (1965) as an estimator of a location parameter, and he established weak consistency. In recent years several authors have examined such estimators using Monte Carlo experiments, but the lack of an asymptotic theory has prevented a more detailed discussion of their properties.

We derive two-sided bounds on the rate of convergence of moments in the central limit theorem. A variety of norming constants is considered, and it is shown that a very delicate alteration to these constants can have a significant effect on the rate. Indeed, the influence of norming constants on rates of convergence of moments is of a more subtle nature than in the central limit theorem itself. We present several examples to illustrate some extreme cases.

Let $\{X_i, i = 1,2, \cdots\}$ be a sequence of positive i.i.d. random variables. Define $S_n = \sum^n_{i=1} X_i$ and $T_n = \sum^n_{i=1} X^2_i$. We study the rate, if any, at which $E\lbrack S^{-2}_n T_n\rbrack \rightarrow 0$.

A function $f(t)$ such that $f(t) / \sqrt{t+1} \uparrow a$ is considered. We define $T = \inf \{t: |W(t)| = f(t)\}$, where $W(t)$ is the Wiener process starting from 0. A sufficient condition for $E\{T^\mu\}$ to be finite is given.

Let $X_1, \cdots, X_n$ be arbitrary random variables and put $S(i, j) = X_i + \cdots + X_j$ and $M(i, j) = \max \{|S(i, i)|, |S(i, i + 1)|, \cdots, |S(i, j)|\}$ for $1 \leq i \leq j \leq n$. Bounds for $E\{\exp tM (1, n)\}, E M^\gamma(1, n)$ and $P\{M(1, n) \geq t\}$ are established in terms of assumed bounds for $E \{\exp t|S(i, j)|\}, E|S(i, j)|^\gamma$ and $P\{|S(i, j)| \geq t\}$, respectively. The bounds explicitly involve a nonnegative function $g(i, j)$ assumed to be quasi-superadditive with index $Q(1 \leq Q \leq 2): g(i, j) + g(j + 1, k) \leq Q g(i, k)$, all $1 \leq i \leq j < k \leq n$. Results previously established for the case $Q = 1$ are improved and are extended to the case $1 < Q < 2$. When $g(i, j)$ is given by $\operatorname{Var} S(i, j)$, applications of the case $Q > 1$ include sequences $\{X_i\}$ exhibiting long-range dependence, in particular certain self-similar processes such as fractional Brownian motion.

A formulation of a Markov chain in a stationary random environment is given, and for the case of finite state space, necessary and sufficient conditions are found under which a state of the chain is inessential, positively essential or properly essential. It is shown that in this case improperly essential states cannot exist.

A simple proof of a central limit theorem for stationary random fields under mixing conditions is given, generalizing some results obtained by more complicated methods, e.g. Bernstein's method.

Some necessary and sufficient conditions for strong laws of large numbers to hold for certain classes of linear functions of order statistics (L.F. of O.S.) are given. The results are used to extend and complement some sufficient conditions for strong laws for L.F. of O.S. derived by van Zwet [5] and Wellner [6]. Also a moment-like condition almost equivalent to the existence of an absolute $p$th moment is introduced.

Let $U_1, U_2 \cdots$ be a sequence of independent uniformly distributed random variables on (0, 1) and $M_n$ be the largest spacing induced by $U_1, \cdots, U_n$. We show that $P(M_n \geq (\log n + 2 \log_2n + \log_3n + \cdots + \log_jn)/n \text{i.o.}) = 1$, where $\log_j$ is the $j$ times iterated logarithm, and $j \geq 4$. If $1 = N_1 < N_2 < \cdots < N_k < \cdots$ is the sequence of the successive times $n$ where $M_n < M_{n-1}$, we derive strong limiting bounds for $\{N_k, k \geq 1\}$.

A result of C. C. Heyde, where it is shown that certain rates of convergence (expressed in terms of series conditions) in the CLT for i.i.d. random variables hold true if and only if corresponding moment conditions are fulfilled, is generalized to $k$-dimensional i.i.d. random vectors.

For a sub-critical Galton-Watson process $\mathbf{X}$ with immigration, estimators have been studied by Pakes (1971) for the mean of the stationary distribution of $\mathbf{X}$, and by Nanthi (1979) for the offspring mean and the immigration mean of $\mathbf{X}$. These estimators have been shown by them to be asymptotically normal. In this paper it is shown that they have a joint limiting distribution which is singularly normal.

Let $X = (X_n; n \geq 0; X_0 = 1)$ be a supercritical Galton-Watson process possessing an offspring mean $1 < m < \infty$, and variance $0 < \sigma^2 < \infty$. The limiting distribution of $\{X^{-1/2}_n(X_{n+r} - \hat{m}^rX_n); r = 2, \cdots, T\}$ where $\hat{m} = X_{n+1}/X_n$, is obtained. As a consequence of this result a Quenouille-Bartlett type of asymptotic goodness of fit test is also proposed for the process $X$.

Let $B$ be the standard one-dimensional Brownian motion, $T_x = \inf\{s \mid s \geq 0; B(s) = x\}, x > 0, T = T_0 \wedge T_a$, \begin{equation*}\gamma_x(t) = \begin{cases} \sup\{s \mid s \leq t; B(s) = x\}\\ , \gamma(t) = \gamma_0(t) \vee \gamma_a(t)\\ 0 \text{if above set} = \phi\end{cases}\end{equation*} where $a > 0$ is fixed and $W(s) = B(T - s), 0 \leq s \leq T$. Let $Z(s)$ be the restriction of $B$ to the interval $\lbrack\gamma(t), t\rbrack$, that is, $Z(s) = B(\gamma(t) + s), 0 \leq s \leq L(t)$. In this paper we use a "time reversal" argument to study relations as $t \rightarrow \infty$ between the processes $Z$ and $W$ under $P^y(\cdot \mid B(t) = x)$ and to evaluate some limits related to $L(t)$.

Let $\varphi$ be a convex function. A sufficient condition is given that a submartingale is equal in law to the $\varphi$-image of a martingale. It follows that each nonnegative submartingale, without any assumption on the regularity of the paths, can be obtained as the absolute value of a martingale.

It is shown that a nonsingular matrix valued continuous semimartingale can be decomposed uniquely as a product of a continuous local martingale and a continuous process of locally bounded variation. An "integration by parts" formula for the multiplicative stochastic integral is also obtained.