On generalized multiplicative cascades

By Quansheng Liu. 2000

207 citations recorded by the source corpus.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/26e29538a947624c160e41339619df3771201058

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In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Asymptotic properties and absolute continuity of laws stable by random weighted mean (2001, methodology): “;|u|=n Au1 : : : Au1 :::un log 1 Au1 : : : Au1 :::un is a martingale, and converges a.s. to a non-trivial solution Z∗¿ 0 of (E) with EZ∗=∞ (see Liu 2000); if EY1 = 1; EY1 log + Y1 = ∞ and E N∑ i=1 Ai logAi¡0; then there is a sequence of positive constants {cn} such that Yn=cn converges in probabilit”
Random walk on sparse random digraphs (2015, background): “A more general version of (7) has been studied by R¨osler [38], Liu [30, 29, 31, 32] and Barral [4, 5].”
A Log-Type Moment Result for Perpetuities and Its Application to Martingales in Supercritical Branching Random Walks (2008, background): “Restricting to the case (A1), the existence of moments of W was studied in quite a number of articles, see [3],[5],[9],[16],[18],[25],[28].”
Discretization Methods for Homogeneous Fragmentations (2004, methodology): “For the branching random walk, it has been considered by Kyprianou [18], Liu [20] with the help of a functional equation and by Biggins and Kyprianou [10] with the measure change method.”
Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in a random environment (2008, methodology): “The model of multiplicative cascades was first introduced by Mandelbrot (1974, [30]); it has been well studied in the literature: see for example Kahane and Peyrière (1976, [25]), Durrett and Liggett (1981,[17]), Guivarc’h (1990, [19]), Franchi (1993, [18] ); for a generalized version and closely re”
Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees (2007, background): “Let us recall from Liu [27] (see also Kyprianou [25]) that, for some c∗ > 0, log 1 W ∗ ∞ law = ξ, (3.”
KPZ formula for log-infinitely divisible multifractal random measures (2008, background): “This model, which arises as the limit of discrete random multipliers, has been the object of numerous studies in probability theory (see for instance [15] for an account on the achieved results).”
Branching random walks (2015, methodology): “Although not mentioned here, it is closely related to multiplicative martingales; see Liu [168], and Harris [124] in the setting of branching Brownian motion.”
Measure change in multitype branching (2004, background): “More relevantly, the convergence in the boundary case has been considered by Kyprianou (1998) and Liu (2000), drawing on results from a related functional equation; the approach here, which is more direct, gives convergence under weaker conditions.”
Elementary fixed points of the BRW smoothing transforms with infinite number of summands (2003, methodology): “To this end, we use the observation that L(Y ) solving (24) is a fixed point of the Poisson shot noise transform (see Iksanov and Yurek (2002) for more details) or, in other words, L(Y ) satisfies (26) and vice versa.”
On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case (2008, methodology): “Applying techniques introduced by Guivarc’h [21] and Liu [ 25 ] and Theorem 1.2 one can prove (of course under appropriate assumptions) that limx→∞xP[Z > x] exists and is positive.”