Let {Xn}n≥1 be a sequence of i.i.d. standard Gaussian random variables, let Sn = ∑n i=1 Xi be the Gaussian random walk, and let Tn = ∑n i=1 Si be the integrated (or iterated) Gaussian random walk. In this paper we derive the following upper and lower bounds for the conditional persistence: P { max 1≤k≤n Tk ≤ 0 ∣∣∣ Tn = 0, Sn = 0} . n, P { max 1≤k≤2n Tk ≤ 0 ∣∣∣ T2n = 0, S2n = 0} & n−1/2 logn , f...

Let {Xi, i = 1, 2, . . .} be i.i.d. standard gaussian variables. Let Sn = X1 + . . . + Xn be the sequence of partial sums and Ln = max 0≤i<j≤n Sj − Si √ j − i . We show that the distribution of Ln, appropriately normalized, converges as n → ∞ to the Gumbel distribution. In some sense, the the random variable Ln, being the maximum of n(n+1)/2 dependent standard gaussian variables, behaves like t...

We study the convergence of densities of a sequence of random variables to a normal density. The random variables considered are nonlinear functionals of a Gaussian process. The tool we are using is the Malliavin calculus, in particular, the integration by parts formula and the Stein’s method. Applications to the convergence of densities of the least square estimator for the drift parameter in ...

1 σ-fields and random variables Week 1 3 1.1 σ-fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 σ-field generated by a collection of events . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Sub-σ-field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Random variables . . . . . . . . ...

We propose the adaptive multicarrier quadrature division–multiuser quadrature allocation (AMQD-MQA) multiple access technique for continuous-variable quantum key distribution (CVQKD). The MQA scheme is based on the AMQD modulation, which granulates the inputs of the users into Gaussian subcarrier continuous-variables (CVs). The subcarrier coherent states formulate Gaussian sub-channels from the...

In this paper, the two parameter ADK entropy, as a generalized of Re'nyi entropy, is considered and some properties of it, are investigated. We will see that the ADK entropy for continuous random variables is invariant under a location and is not invariant under a scale transformation of the random variable. Furthermore, the joint ADK entropy, conditional ADK entropy, and chain rule of this ent...

We consider quadratic forms Q n = X 1j<kn a jk X j X k ; where X j are i.i.d. random variables with nite third moment. We obtain optimal bounds for the Kolmogorov distance between the distribution of Q n and the distribution of the same quadratic forms with X j replaced by corresponding Gaussian random variables.

We consider the generalized differential entropy of normalized sums of independent and identically distributed (IID) continuous random variables. We prove that the Rényi entropy and Tsallis entropy of order α (α > 0) of the normalized sum of IID continuous random variables with bounded moments are convergent to the corresponding Rényi entropy and Tsallis entropy of the Gaussian limit, and obtai...

We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.