By D. Kannan
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This paintings indicates glossy probabilistic tools in motion: Brownian movement procedure as utilized to phenomena invesitigated via eco-friendly et al. It starts with the Newton-Coulomb strength and ends with options via first and final exits of Brownian paths and conductors.
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H(z) = cle -bz Hence , b > 0 , from which F(z) kCl = 1-G(z) = 1 - - - e But s i n c e m = E ( x k ) , F(0+) = 0. 3. (X-u) + = max(X-u,0). , b > 0, z ~ 0 . (34) w i t h z = 0 and Len~rna 1 . 2 . 1 Consequently, Let us r e t u r n interesting -bz m imply that G(0+) = 1 o r (kCl)/m~l , which completes the proof. to the linear case. of the exponential D K-M. Chong (1977) points out the following distribution. Let X ~ 0 be a random variable with finite expectation. Let If the distribution of X is continuous at zero, then X has a negative exponential distribution if, and only if, (38) E[(X-s)+]E[(X-t) +] = mE[(X-s-t) +] , all s,t ~> 0 , with some constant m > 0.
2. G(x) = l-F(x). Let F(x) be a nondegenerate distribution function and put Assume that (3) holds for two integral values of n, n I and n2, say, such that (log nl)/(log n2) is irrational, arbitrary. =x n -- x ~ 0 is Then F(x) = l-e -bx, x ~ 0, with some b > 0. This result is due to J. Sethuraman (1965). His result is actually more gen- eral than the statement above; the more general case will be discussed in subsequent chapters. 2 was later reobtained by B. Arnold ~1971) (assuming n 2 = nl+l), whose proof is different from the one presented below.
G. Laurent (initiated by him in the 1950's, but for convenience we refer to his published paper of 1974), is given below. N. Vartak (1974). 1. 0 -< y < +~. Let g(y) > 0 be a decreasing and differentiable function for Assume further that g'(y) _> -i with g'(0) ~ -i and such that (26) Af l+g' g(y)(y) dy 0 is finite or infinite according as A is finite or infinite. If X is a nonnegative random variable with continuous distribution F(x) and such that (27) E(X-ylX_>y) = g(y) , all y _> 0 , then the distribution F(x) is uniquely determined.
An introduction to stochastic processes by D. Kannan