The reversed ladder of a random walk

1977 ◽  
Vol 14 (1) ◽  
pp. 190-194 ◽  
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Usual Sense ◽  
Random Vectors ◽  
Ladder Height ◽  
Ladder Epoch

The reversed strict ascending ladder epoch v1 + ··· +vm of the random walk S(n) with drift to ∞ is the m th time n at which the event S(n) < S(k), k > n, occurs and the corresponding ladder height is H1 + ··· + Hm = S(v1 + ··· + vm). It is shown that the random vectors (vi, Hi) are independent and for i ≧ 2 have the same distribution as the first strict ascending ladder time and height in the usual sense. This leads to an equality between the distribution of the first strict ladder height and P{min S(n) ≧ x} for x > 0. Reversed weak ladder points are defined analogously.

The reversed ladder of a random walk

1977 ◽  
Vol 14 (01) ◽  
pp. 190-194
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Usual Sense ◽  
Random Vectors ◽  
Ladder Height ◽  
Ladder Epoch

The reversed strict ascending ladder epoch v 1 + ··· +vm of the random walk S(n) with drift to ∞ is the m th time n at which the event S(n) &lt; S(k), k &gt; n, occurs and the corresponding ladder height is H 1 + ··· + Hm = S(v 1 + ··· + vm ). It is shown that the random vectors (vi, Hi ) are independent and for i ≧ 2 have the same distribution as the first strict ascending ladder time and height in the usual sense. This leads to an equality between the distribution of the first strict ladder height and P{min S(n) ≧ x} for x &gt; 0. Reversed weak ladder points are defined analogously.

Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (4) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

Moments for multidimensional Mandelbrot cascades

2016 ◽  
Vol 53 (1) ◽  
pp. 1-21
Author(s):  
Chunmao Huang
Keyword(s):  
Random Walk ◽  
Random Matrices ◽  
Random Variable ◽  
Tail Probability ◽  
Decay Rates ◽  
Branching Random Walk ◽  
Sufficient Condition ◽  
Random Vectors ◽  
Products Of Random Matrices ◽  
Sums Of Products

Abstract We consider the distributional equation Z =D ∑k=1NAkZ(k), where N is a random variable taking value in N0 = {0, 1, . . .}, A1, A2, . . . are p x p nonnegative random matrices, and Z, Z(1), Z(2), . . ., are independent and identically distributed random vectors in R+p with R+ = [0, ∞), which are independent of (N, A1, A2, . . .). Let {Yn} be the multidimensional Mandelbrot martingale defined as sums of products of random matrices indexed by nodes of a Galton–Watson tree plus an appropriate vector. Its limit Y is a solution of the equation above. For α > 1, we show a sufficient condition for E|Y|α ∈ (0, ∞). Then for a nondegenerate solution Z of the distributional equation above, we show the decay rates of Ee-t∙Z as |t| → ∞ and those of the tail probability P(y ∙ Z ≤ x) as x → 0 for given y = (y1, . . ., yp) ∈ R+p, and the existence of the harmonic moments of y ∙ Z. As an application, these results concerning the moments (of positive and negative orders) of Y are applied to a special multitype branching random walk. Moreover, for the case where all the vectors and matrices of the equation above are complex, a sufficient condition for the Lα convergence and the αth-moment of the Mandelbrot martingale {Yn} are also established.

Understanding the Wiener–Hopf factorization for the simple random walk

1994 ◽  
Vol 31 (2) ◽  
pp. 561-563 ◽  
Author(s):  
Joanne Kennedy
Keyword(s):  
Random Walk ◽  
Sample Path ◽  
Direct Proof ◽  
Simple Random Walk ◽  
Ladder Height

We give a sample path proof of the well-known Wiener–Hopf identity F = G– + G+ – G– ∗G+ which relates the ladder-height distributions G– and G+ of a simple random walk to the step distribution F. Unlike previous approaches this direct proof is both simple and intuitive.

scholarly journals Asymptotics for the moments of the overshoot and undershoot of a random walk

2009 ◽  
Vol 41 (2) ◽  
pp. 469-494 ◽  
Author(s):  
Zhaolei Cui ◽  
Yuebao Wang ◽  
Kaiyong Wang
Keyword(s):  
Random Walk ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Strong Markov Property ◽  
Renewal Equations ◽  
Ladder Height ◽  
Overshoot And Undershoot ◽  
Equivalent Conditions ◽  
Strong Markov

In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

On the local behaviour of ladder height distributions

1994 ◽  
Vol 31 (03) ◽  
pp. 816-821 ◽  
Author(s):  
J. Bertoin ◽  
R. A. Doney
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Mass Function ◽  
Height Distribution ◽  
Local Behaviour ◽  
Ladder Height ◽  
Asymptotic Behaviours

There is a well-known connection between the asymptotic behaviour of the tail of the distribution of the increasing ladder height and the integrated tail of the step distribution of a random walk which either drifts to –∞, or oscillates and whose decreasing ladder height has finite mean. We establish a similar connection in a local sense; this means that in the lattice case we link the asymptotic behaviours of the mass function of the ladder height distribution and of the tail of the step distribution. We deduce the asymptotic behaviour of the mass function of the maximum of the walk, when this is finite, and also treat the non-lattice case.

scholarly journals Asymptotics of the density of the supremum of a random walk with heavy-tailed increments

2006 ◽  
Vol 43 (03) ◽  
pp. 874-879 ◽  
Author(s):  
Yuebao Wang ◽  
Kaiyong Wang
Keyword(s):  
Random Walk ◽  
Ladder Height ◽  
Heavy Tailed ◽  
Equivalent Conditions

Under some relaxed conditions, in this paper we obtain some equivalent conditions on the asymptotics of the density of the supremum of a random walk with heavy-tailed increments. To do this, we investigate the asymptotics of the first ascending ladder height of a random walk with heavy-tailed increments. The results obtained improve and extend the corresponding classical results.

Corrected diffusion approximations in certain random walk problems

1979 ◽  
Vol 11 (04) ◽  
pp. 701-719 ◽  
Author(s):  
D. Siegmund
Keyword(s):  
Random Walk ◽  
Diffusion Approximation ◽  
Diffusion Approximations ◽  
Infinite Time ◽  
Ladder Height ◽  
Correction Terms

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

scholarly journals Asymptotics of the density of the supremum of a random walk with heavy-tailed increments

2006 ◽  
Vol 43 (3) ◽  
pp. 874-879 ◽  
Author(s):  
Yuebao Wang ◽  
Kaiyong Wang
Keyword(s):  
Random Walk ◽  
Ladder Height ◽  
Heavy Tailed ◽  
Equivalent Conditions

Under some relaxed conditions, in this paper we obtain some equivalent conditions on the asymptotics of the density of the supremum of a random walk with heavy-tailed increments. To do this, we investigate the asymptotics of the first ascending ladder height of a random walk with heavy-tailed increments. The results obtained improve and extend the corresponding classical results.

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