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Define a certain gambler’s ruin process \(\mathbf {X}_{j}, \, j\ge 0,\) such that the increments \(\varepsilon _{j}:=\mathbf {X}_{j}-\mathbf {X}_{j-1}\) take values \(\pm 1\) and satisfy \(P(\varepsilon _{j+1}=1|\varepsilon _{j}=1, |\mathbf {X}_{j}|=k)=P(\varepsilon _{j+1}=-1|\varepsilon _{j}=-1,|\mathbf {X}_{j}|=k)=a_k\), all \(j\ge 1\), where \(a_k=a\) if \( 0\le k\le f-1\), and \(a_k=b\) if \(f\le k

Let X j denote a fair gambler’s ruin process on Z ∩ [ − N , N ] started from X 0 = 0 , and denote by R N the number of runs of the absolute value, | X j | , until the last visit j = L N by X j to 0. Then, as N → ∞ , N − 2 R N converges in distribution to a density with Laplace transform: tanh ( λ ) / λ . In law, we find: 2 ( lim N → ∞ N − 2 R N ) = lim N → ∞ N − 2 L N . Denote by R N ′ and L N ′ the number of runs and steps respectively in the meander portion of the gambler’s ruin process. Then, N − 1 ( 2 R N ′ − L N ′ ) converges in law as N → ∞ to the density ( π / 4 ) sech 2 ( π x / 2 ) , − ∞ x ∞ .

We prove that the expected number of particles of the critical contact process on a homogeneous tree is bounded above. This is the first graph for which the behavior of the expected number of particles of the critical contact process is known. As an easy corollary of our result we get that the critical contact process dies out on any homogeneous tree. This completes the work of Pemantle (1992).

Let $\{X_n\} = \{X^{(N)}_n\}$ be a Markov chain in the probability measures $\mathbf{P}\lbrack 0, 1\rbrack$, equipped with a certain metric for the topology of weak convergence, and denote $\mathbf{E}_x(X_1) = \mathbf{f}_N(x)$. Define a projection $\xi = \pi_dx$ on $\mathbf{P}\lbrack 0, 1\rbrack$ by defining $\xi$ to be absolutely continuous with respect to Lebesgue measure with constant density $(d + 1)x(A)$ on each interval $A$ of an equipartition of $\lbrack 0, 1 \rbrack$ into $d + 1$ intervals, and let $\pi_d\mathbf{P}\lbrack 0, 1\rbrack \subset \mathbb{R}^d$ be the natural embedding. Assume $\pi_d\mathbf{f}_N(\xi) \underline{\sim} \xi + \beta_N h_d(\xi), h_d(\xi_{0, d}) = \mathbf{0}$ and $\operatorname{Cov}_\xi(\pi_d X_1) \underline{\sim} \sigma^2_d(\xi)/N$, in certain senses as $N \rightarrow \infty$, where $\xi_{0, d}$ is an asymptotically stable fixed point of $\pi_d\mathbf{f}_N(\xi) = \xi$ and $x_0 = \lim_{d \rightarrow \infty} \xi_{0, d}$ exists in $\mathbf{P}\lbrack 0, 1\rbrack$. Assuming various regularity conditions and $\beta = \beta_N \rightarrow 0, N\beta/\log N \rightarrow \infty$, it is shown that the expected time it takes the Markov chain to exit a fixed open ball $D$ about $x_0$ once $X_0 \in D$ is logarithmically equivalent to $\exp\lbrack N\beta_N V \rbrack$, where $V > 0$ is a limit of solutions $V_d = V_d(h_d, \sigma_d)$ of variational problems of Wentzell-Freidlin type in $\mathbb{R}^d$ as $d \rightarrow \infty$. These results apply to an infinite alleles model in population genetics, where $\{X_n\}$ represents the evolution of distributions of types among a population of $N$ randomly mating genes, and where forces of mutation and selection are stronger than effects due to finite population size.

We obtain an almost sure approximation of a martingale with values in a real separable Hilbert space H H by a suitable H H -valued Brownian motion.

A variant of Doob’s maximal inequality is obtained for dyadic martingales with multiple indices. The inequality furnishes a precise estimate of the L p {L^p} norm of the maximal function in terms of the L p {L^p} norms of the jumps, p ≥ 2 p \geq 2 .

Given independent identically distributed random variables {xn;n ∈ ℕq| indexed by q-tuples of positive integers and taking values in a separable Banach space B we approximate the rectangular sums \(\{ \sum\limits_{m \leqq n} {x_m ;n \in \mathbb{N}^q } \} \) by a Brownian sheet. We obtain the corresponding result for random variables with values in a separable Hilbert space H while assuming an optimal moment condition. Generalized versions of the functional law of the iterated logarithm are thus derived.

The central limit theorem for stationary linearly dependent sequences is extended for elements in the space of continuous functions on a compact metric space. The proof is based on a new estimate for exponential-type moments of sums of independent random variables.

Given a random field {ξν, ν∈Z + q } indexed by q-tuples of positive integers and satisfying a strong mixing condition we study the approximation of the partial sum field {Sn, n∈Z + q} by Brownian sheet. Setting $$G_\alpha = \{ (n_1 ,...,n_q ) \in Z_ + ^q :n_k \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } (\mathop \Pi \limits_{1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } q,i \pm k} n_i )^\alpha , k = 1,...,q\}$$ for 00. We also construct an example showing that in a somewhat larger, similar type domain the above approximation is generally impossible, even with λ=0.

Let |xj,j∈ℕq} be independent identically formed random elements with values in a Banach spaceB, indexed byq-tuples of positive integers. We obtain the almost sure approximation as well as the approximation in probability of the partial sum process\(\{ \sum\limits_{j \in nA} {x_j ,A \in A\} } \) by a partial sum process\(\{ \sum\limits_{j \in nA} {y_j ,A \in A\} } \) asn→∞ uniformly over all setsA in a certain class\(A\) of subsets of theq-dimensional unit cube. Here |yj,j∈ℕq} are independent identically distributed Gaussian random variables. These results are then applied to obtain the approximation of empirical processes over sets and indexed by sets by Gaussian partial sum processes indexed by sets.

The infinite dimensional Ornstein-Uhlenbeck process is derived as the weak limit of processes xn(t,s) constructed from first hitting position of spheres sn−1(et/2) for standard Brownian motion in Rn starting at the origin.

Let $\{X_n\}$ be a Markov chain on a bounded set in $R^d$ with $E_x(X_1) = f_N(x) = x + \beta_N h_N(x)$, where $x_0$ is a stable fixed point of $f_N(x) = x$, and $\operatorname{Cov}_x(X_1) \approx \sigma^2(x)/N$ in various senses. Let $D$ be an open set containing $x_0$, and assume $h_N(x) \rightarrow h(x)$ uniformly in $D$ and either $\beta_N \equiv 1$ or $\beta_N \rightarrow 0, \beta_N \gg \sqrt{\log N/N}$. Then, assuming various regularity conditions and $X_0 \in D$, the time the process takes to exit from $D$ is logarithmically equivalent in probability to $e^{VN\beta_N}$, where $V > 0$ is the solution of a variational problem of Freidlin-Wentzell type $\lbrack \text{if} \beta_N \rightarrow 0 \text{and} d = 1, V = \inf\{2 \int^y_{x_0}\sigma^{-2}(u)|h(u) du|: y \in \partial D\} \rbrack$. These results apply to the Wright-Fisher model in population genetics, where $\{X_n\}$ represent gene frequencies and the average effect of forces such as selection and mutation are much stronger than effects due to finite population size.