It is well known that the distribution of simple random walks on ℤ conditioned on returning to the origin after 2n steps does not depend on p=P(S1=1), the probability of moving to the right. Moreover, conditioned on {S2n=0} the maximal displacement maxk≤2n|Sk| converges in distribution when scaled by √n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on ℤ. We show that under the quenched law Pω (conditioned on the environment ω), the maximal...

A simple personal saving model with interest rate based on random fluctuation of national growth rate is considered. We establish connections between the mean stochastic stability of our model and the deterministic stability of related partial difference equations. Then the asymptotic behavior of our stochastic model is studied. Although the model is simple, the techniques for obtaining its properties are not, and we make use of the theory of abstract Banach algebras and weighted spaces. It is hoped...

In the present paper we prove moderate deviations for a Curie–Weiss model with external magnetic field generated by a dynamical system, as introduced by Dombry and Guillotin-Plantard in [C. Dombry and N. Guillotin-Plantard, Markov Process. Related Fields 15 (2009) 1–30]. The results extend those already obtained for the Curie–Weiss model without external field by Eichelsbacher and Löwe in [P. Eichelsbacher and M. Löwe, Markov Process. Related Fields 10 (2004) 345–366]. The Curie–Weiss model with...

We derive necessary and sufficient conditions for a sum of i.i.d. random variables ${\sum }_{i=1}^n{X}_i/b_n$ – where $\frac{b_n}{n}\downarrow 0$, but $\frac{b_n}{\sqrt{n}}\uparrow \infty$ – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

We derive necessary and sufficient conditions for a sum of i.i.d. random variables ${\sum }_{i=1}^n{X}_i/b_n$ – where $\frac{b_n}{n}\downarrow 0$, but $\frac{b_n}{\sqrt{n}}\uparrow \infty$ – to satisfy a moderate deviations principle. Moreover we show that this equivalence is a typical moderate deviations phenomenon. It is not true in a large deviations regime.

Let X1,...,Xn1 be a random sample from a population with mean µ1 and variance ${\sigma }_1^2$, and X1,...,Xn1 be a random sample from another population with mean µ2 and variance ${\sigma }_2^2$ independent of {Xi,1 ≤ i ≤ n1}. Consider the two sample t-statistic $T=\frac{\overline{X}-\overline{Y}-({\mu }_1-{\mu }_2)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}$. This paper shows that ln P(T ≥ x) ~ -x²/2 for any x := x(n1,n2) satisfying x → ∞, x = o(n1 + n2)1/2 as n1,n2 → ∞ provided 0 < c1 ≤ n1/n2 ≤ c2 < ∞. If, in addition, E|X1|3 < ∞, E|Y1|3 < ∞, then $\frac{P(T\ge x)}{1-\Phi \left(x\right)}\to 1$ holds uniformly in x ∈ (O,o((n1 + n2)1/6))

This paper gives upper and lower bounds for moments of sums of independent random variables $\left(X_k\right)$ which satisfy the condition ${P\left(\right|X|}_k\ge t)=exp(-N_k\left(t\right))$, where $N_k$ are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which $N\left(t\right)={\left|t\right|}^r$ for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapień who have done the same for convex functions N.

Mathematics Subject Classification: 65C05, 60G50, 39A10, 92C37In this paper the multi-dimensional Monte-Carlo random walk simulation models governed by distributed fractional order differential equations (DODEs) and multi-term fractional order differential equations are constructed. The construction is based on the discretization leading to a generalized difference scheme (containing a finite number of terms in the time step and infinite number of terms in the space step) of the Cauchy problem for...

Nous introduisons une notion de moyenne harmonique pour une marche aléatoire sur une relation d’équivalence mesurée graphée, qui généralise la notion classique de moyenne invariante. Pour les graphages à géométrie bornée, une telle moyenne existe toujours. Nous prouvons qu’une moyenne harmonique devient invariante lorsque la marche aléatoire sur presque toute orbite jouit de bonnes propriétés asymptotiques telles que la propriété de Liouville ou la récurrence.

Let $\{X_n,n\ge 1\}$ be a sequence of independent random variables such that $E{X}_n=a_n$, $E{(X_n-a_n)}^2={\sigma }_n^2$, $n\ge 1$. Let $\{N_n,n\ge 1\}$ be a sequence od positive integer-valued random variables. Let us put $S_{N_n}={\sum }_{k=1}^{N_n}{X}_k$, $L_n={\sum }_{k=1}^n{a}_k$, $s_n^2={\sum }_{k=1}^n{\sigma }_k^2$, $n\ge 1$. In this paper we present necessary and sufficient conditions for weak convergence of the sequence $\{(S_{N_n}-L_n)/s_n,n\ge 1\}$, as $n\to \infty$. The obtained theorems extend the main result of M. Finkelstein and H.G. Tucker (1989).