Nina Gantert

Bio: Nina Gantert is an academic researcher from Technische Universität München. The author has contributed to research in topics: Random walk & Branching random walk. The author has an hindex of 20, co-authored 112 publications receiving 1548 citations. Previous affiliations of Nina Gantert include University of Bonn & Technical University of Berlin.

Quenched, annealed and functional large deviations for one-dimensional random walk in random environment

Abstract: Suppose that the integers are assigned random variables {ω i} (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X n} (called a RWRE) which, when at i, moves one step to the right with probability ω i, and one step to the left with probability 1 − ω i. When the {ω i} sequence is i.i.d., Greven and den Hollander (1994) proved a large deviation principle for X n/n, conditional upon the environment, with deterministic rate function.We consider in this paper large deviations, both conditioned on the environment (quenched) and averaged on the environment (annealed), for the RWRE, in the ergodic environment case. The annealed rate function is the solution of a variational problem involving the quenched rate function and specific relative entropy. We also give a detailed qualitative description of the resulting rate functions. Our techniques differ from those of Greven and den Hollander, and allow us to present also a trajectorial (quenched) large deviation principle.

The speed of biased random walk on percolation clusters

Abstract: We consider biased random walk on supercritical percolation clusters in ℤ2. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive.

The speed of biased random walk on percolation clusters

Abstract: We consider biased random walk on supercritical percolation clusters in $\Z^2$. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive.

Biased random walks on Galton–Watson trees with leaves

Abstract: We consider a biased random walk Xn on a Galton–Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending on the bias β, such that |Xn| is of order nγ. Denoting Δn the hitting time of level n, we prove that Δn/n1/γ is tight. Moreover, we show that Δn/n1/γ does not converge in law (at least for large values of β). We prove that along the sequences nλ(k) = ⌊λβγk⌋, Δn/n1/γ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton–Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Methods Of Modern Mathematical Physics

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