Kai Lai Chung

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  • Kai Lai Chung
  • Proceedings of the National Academy of Sciences…
  • 1947
this becomes false if (BI), (Be) and (B) are replaced by (Nt), (NM) and (N), respectively. This follows, even for p = 2 = q, from the above example proving that (NW) is not linear. Correspondingly, (Ne) cannot be interpreted as the dual space of (NP), since such an interpretation would involve the definition of a scalar product. 7. Let (Nt) denote the space(More)
2. S. I. Grossman and R. K. Miller, Nonlinear Volterra integro-differential systems with L kernels, J. Differential Equations 13 (1973), 551-566. 3. M. J. Leitman and V. J. Mizel, Hereaitary laws and nonlinear integral equations on the line, J. Differential Equations (to appear). 4. J. L. Levin and J. A. Nohel, Perturbations o f a nonlinear Volterra(More)
Let X(t) (05 t < 00) be the Brownian motion process. Concerning the uniform continuity of X(t), there exists P. Levy's result. Before stating his result, let us define the concept of upper class and lower class with regard to the uniform continuity of X(t) (0 2 ts 1). If there exists a positive number E such that 1 t'-t[ 5 E implies the relation (1)(More)
© Springer-Verlag, Berlin Heidelberg New York, 1980, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une(More)
and computed the second moment of the difference of the expressions in (1) and (2). In this paper, by examining the excursions in Brownian motion and using a new formula for the distribution of their maxima, we obtain a direct identification of the limit in (1) without using (2). Let Tx=inf{t: W(t )=x} , Tx=inf{t: Y( t )=x} . For a > 0 let R~=0, R~= T/, +(More)
© Springer-Verlag, Berlin Heidelberg New York, 1985, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une(More)
paper we study the question : if u , is finite, can we enlarge D to a domain G so that u G is still finite? First, we prove that we can always add a finite n u m b e r of balls centered on ~D to get such a G. But as we add more and more such balls, their radii may have to shrink to zero so that it is not always possible to cover the entire bounda ry of D.(More)
The celebrated Borel-Cantelli lemma asserts that (A) If ~P(E,,) < 00, then P (lim sup Ek) =O; (B) If the events Es are independent and if xP(Ek) = m, then P(lim sup Eh) = 1. In intuitive language P(lim. sup &) is the probability that the events Eh occur “infinitely often” and will be denoted by P(Ek i.0.). This lemma is the basis of all theorems of the(More)