Strassen's Law Research Articles

Strassen's law for local time

Strassen's law for local time

Another Version of Strassen's Log Log Law with an Application to Approximate Upper Functions of a Gaussian Process with a Positive Index

Let $\{Y(t, \omega) = (X_1(t, \omega), \cdots, X_d(t, \omega)); 0 \leq t \leq 1\}$ be a $d$-dimensional Gaussian process whose components are independent copies of a Gaussian process with index $\alpha$; that is, $E\lbrack X(t, \omega)\rbrack = 0, X(0, \omega) = 0$, and $E\lbrack(X(t, \omega) - X(s, \omega))^2\rbrack = \sigma^2(|t - s|)$, where $\sigma(t) = t^\alpha, 0 < \alpha < 1$. Let $h(t)$ be a positive, non-increasing, continuous function and set $q = \sup\big\{r \geq 0; \int_{+0} e^{-rh^2(t)/2} dt/t = + \infty\big\}.$ Then, as an application of a version of Strassen's $\log \log$ law, we have \begin{equation*}\begin{split}\lim \sup_{t\downarrow 0} t^{-1}m(\{0 \leq s \leq t;\|Y(s, \omega)\| > \sigma(s)h(s)\}) \\ &= \sup_{x\in B}m(\{0 \leq s \leq 1;\|x(s)\| \geq \sigma(s)/ \sqrt{q}\}), \quad\text{a.s.},\end{split}\end{equation*} where $\| \|$ denotes the usual Euclidean norm, $m(\Gamma)$ denotes the Lebesgue measure of a linear set $\Gamma$, and $B$ is the unit ball of the direct sum of $d$ copies of the reproducing kernel Hilbert space with the kernel $R(s, t) = (\sigma^2(t) + \sigma^2(s) - \sigma^2(|t - s|))/2$. In case of the $d$-dimensional Brownian motion, Strassen [7] had proved that the right-hand side of the above formula is equal to $1 - \exp\{-4(q - 1)\}$ if $q \geq 1$, and 0 if $q \leq 1$. As a corollary, $\sigma(t)h(t)$ is an approximate upper function as introduced by D. German [2] if and only if $q \leq 1$. Especially, if $\lim_{t\downarrow0}h(t)/ \sqrt{2 \log \log 1/t} = c, \sigma(t)h(t)$ is an approximate upper function if and only if $c \geq 1$.

On Strassen's law of the iterated logarithm

Kolmogorov's law of the iterated logarithm has been sharpened by Strassen who proved a more refined theorem by using tools from functional analysis. The present paper gives a “classical” proof of Strassen's theorem, using a method along the lines of Kolmogorov's original approach. At the same time the result proved here is more general since a) the random variables involved need not have the same distributions, b) the condition of independence is weakened and c) instead of Kolmogorov's growth condition on the random variables, only a mild restriction on their moments of order l≧3 is needed.

A unification of Strassen's law and L�vy's modulus of continuity

A unification of Strassen's law and L�vy's modulus of continuity

A relation between Chung's and Strassen's laws of the iterated logarithm

Let W(t) be a standard Wiener process and let f(x) be a function from the compact class in Strassen's law of the iterated logarithm. We investigate the lim inf behavior of the variable sup ¦W(xT)(2T loglog T)−1/2−f(x)¦, 0≦x≦1 suitably normalized as T→∞.

A generalization of Strassen's functional law of iterated logarithm

Let {a T; T≧0} be a non-decreasing function of T with 0<a T≦T and let {W(t); t≧0} be a Wiener process. The limit functions of the processes $$\Gamma _{t,T} (x) = \beta _T (W(t + xa_T ) - W(t)) (0\underline \leqslant x \underline \leqslant 1)$$ where 0≦t≦T−a T, T>0 and $$\beta _T = (2a_T (\log Ta_T^{ - 1} + \log \log T))^{ - \tfrac{1}{2}} $$ are characterized. In the case a T=T Strassen's law is obtained as a special case.

On the Speed of Convergence in Strassen's Law of the Iterated Logarithm

Here there is derived a condition on sequences $\varepsilon_n \downarrow 0$ which implies that $P\lbrack W(n^\bullet)/(2n \log \log n)^\frac{1}{2} \not\in K^\varepsilon n \mathrm{i.o.}\rbrack = 0$, where $W$ is the Wiener process and $K$ is the compact set in Strassen's law of the iterated logarithm. A similar result for random walks is also given.

Strassen's law of the iterated logarithm

Il s’agit d’établir la forme fonctionnelle de Strassen de la loi du logarithme itéré pour les sommes partielles de variables aléatoires à valeurs dans la limite inductive stricte d’espaces de Fréchet, qui sont de type d’espace d’Hilbert. La démonstration dépend de l’obtention des estimations de Barry-Esssen pour les variables aléatoires à valeurs dans un espace d’Hilbert.

The Law of the Iterated Logarithm for Gaussian Processes

Strassen's law of the iterated logarithm for Brownian motion is extended to a class of Gaussian processes. Let $\{X(t), t \geqq 0\}$ be a real continuous Gaussian process with $X(0) = 0$, mean zero and continuous covariance kernel $R(s, t)$. Define a random sequence $\{f_n(t, \omega)\}$ in $C\lbrack 0, 1 \rbrack$ by $f_n(t, \omega) = X(nt, \omega)/(2R(n, n) \log \log n)^{\frac{1}{2}}$. Under certain conditions on $R$ it is shown that with probability one $\{f_n(t, \omega)\}$ is equicontinuous and the set of its limit points is the unit ball of the reproducing kernel Hilbert space with reproducing kernel $\Gamma$ determined by $R$. The result generalizes the author's earlier result (1972).

Some results for probability measures on linear topological vector spaces with an application to Strassen's log log law

We prove some results regarding tight probability measures on real Frechet spaces and countable strict inductive limits of these spaces. These results are applied to Gaussian measures and to construct a Brownian motion on such spaces. We then prove the log log law of Strassen for this Brownian motion.