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1.
Extremes - We consider a random censoring model for survival analysis, allowing the possibility that only a proportion of individuals in the population are susceptible to death or failure, and the... 相似文献
2.
We prove some limiting results for a Lévy process X
t
as t0 or t, with a view to their ultimate application in boundary crossing problems for continuous time processes. In the present paper we are mostly concerned with ideas related to relative stability and attraction to the normal distribution on the one hand and divergence to large values of the Lévy process on the other. The aim is to find analytical conditions for these kinds of behaviour which are in terms of the characteristics of the process, rather than its distribution. Some surprising results occur, especially for the case t0; for example, we may have X
t
/t
P
+ (t0) (weak divergence to +), whereas X
t
/t a.s. (t0) is impossible (both are possible when t), and the former can occur when the negative Lévy spectral component dominates the positive, in a certain sense. Almost sure stability of X
t
, i.e., X
t
tending to a nonzero constant a.s. as t or as t0, after normalisation by a non-stochastic measurable function, reduces to the same type of convergence but with normalisation by t, thus is equivalent to strong law behaviour. Boundary crossing problems which are amenable to the methods we develop arise in areas such as sequential analysis and option pricing problems in finance. 相似文献
3.
Harry Kesten R. A. Maller 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》1999,35(6):685
We show that the passage time, T*(r), of a random walk Sn above a horizontal boundary at r (r≥0) is stable (in probability) in the sense that
as r→∞ for a deterministic function C(r)>0, if and only if the random walk is relatively stable in the sense that
as n→∞ for a deterministic sequence Bn>0. The stability of a passage time is an important ingredient in some proofs in sequential analysis, where it arises during applications of Anscombe's Theorem. We also prove a counterpart for the almost sure stability of T*(r), which we show is equivalent to E|X|<∞, EX>0. Similarly, counterparts for the exit of the random walk from the strip {|y|≤r} are proved. The conditions arefurther related to the relative stability of the maximal sum and the maximum modulus of the sums. Another result shows that the exit position of the random walk outside the boundaries at ±r drifts to ∞ as r→∞ if and only if the random walk drifts to ∞. 相似文献
4.
R.A. Maller 《Stochastic Processes and their Applications》1980,10(1):65-73
We show that if Xi is a stationary sequence for which converges to a finite non zero random variable of constant sign, where Sn=X1+X2+?+Xn and Bn is a sequence of constants, then Bn is regularly varying with index 1. If in addition is finite, then is finite, and if in addition to this Xi satisfies an asymptotic independence condition, . 相似文献
5.
Cell decompositions are constructed for polynomials f(x)Zp[x] of degree n, such that n<p, using O(n2) cells. When f is square-free this yields a polynomial-time algorithm for counting and approximating roots in Zp. These results extend to give a polynomial-time algorithm in the bit model for fZ[x]. 相似文献
6.
We give necessary and sufficient conditions for the almost sure relative stability of the overshoot of a random walk when
it exits from a two-sided symmetric region with curved boundaries. The boundaries are of power-law type, ±rn
b
, r > 0, n = 1, 2,..., where 0 ≤ b < 1, b≠ 1/2. In these cases, the a.s. stability occurs if and only if the mean step length of the random walk is finite and non-zero,
or the step length has a finite variance and mean zero.
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7.
Maller V. N. Naidu M. S. 《IEEE transactions on plasma science. IEEE Nuclear and Plasma Sciences Society》1975,3(2):49-54
The sparking potentials and swarm coefficients (ionization and attachment coefficients) have been measured in sulphurhexafluoride- air and freon-nitrogen mixtures over the range of 110 ? E/p ? 240 V cm-1 torr-l and gas pressures varying between 1 and 20 torr, at 20°C. Addition of strongly attaching salphur-hexafluoride and freon gases increased the sparking potentials and the rate of increase of the attachment coefficient with increasing percentage of the strongly attaching gases in the mixtures was much larger than the rate of change of the first ionization coefficient. 相似文献
8.
Summary. Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S
n
into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S
n
→∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the
increments of S
n
, are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite.
A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which
the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.
Received: 18 September 1995/In revised form: 28 February 1996 相似文献
9.
We prove necessary and sufficient conditions for the almostsure convergence of the integrals
and thus of ,where Mt = sup{|Xs|: s t} is the two-sided maximum processcorresponding to a Lévy process (Xt)t 0, a(·)is a non-decreasing function on [0, ) with a(0) = 0, g(·)is a positive non-increasing function on (0, ), possibly withg(0 + ) = , and f(·) is a positive non-decreasing functionon [0, ) with f(0) = 0. The conditions are expressed in termsof the canonical measure, (·), of the process Xt. Thespecial case when a(x) = 0, f(x) = x and g(·) is equivalentto the tail of (at zero or infinity) leads to an interestingcomparison of Mt with the largest jump of Xt in (0, t]. Some results concerning the convergence at zero and infinityof integrals like t g(a(t) + |Xt|) dt, t g(St) dt,and t g(Rt) dt, where St is the supremum process and Rt= St Xt is the process reflected in its supremum, arealso given. We also consider the convergence of integrals suchas , etc. 相似文献
10.