Large deviations for trajectories of sums of independent random variables

Given a sequence of independent, but not necessarily identically distributed random variables,Y _i , letS _k denote thekth partial sum. Define a function by taking to be the piecewise linear interpolant of the points (k, S _k ), evaluated att, whereS ₀=0, andk=0, 1, 2,... Fort[0, 1], let . The are called trajectories. With regularity and moment conditions on theY _i , a large deviation principle is proved for the .     

Integro-local and integral theorems for sums of random variables with semiexponential distributions

We obtain some integro-local and integral limit theorems for the sums S(n) = ξ(1) + ? + ξ(n) of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form $P(\xi \ge t) = e^{ - t^\beta L(t)} $ , where β ∈ (0, 1) and L(t) is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as x → ∞ of the probabilities P(S(n) ∈ [x, x + Δ)) and P(S(n) ≥ x) in the zone of normal deviations and all zones of large deviations of x: in the Cramér and intermediate zones, and also in the “extreme” zone where the distribution of S(n) is approximated by that of the maximal summand.     

Superlarge deviation probabilities for sums of independent lattice random variables with exponential decreasing tails

In the note we study large and superlarge deviation probabilities of sum of i.i.d. lattice random variables, whose distribution function has an exponentially decreasing tail at infinity.     

Moderate deviations principle for products of sums of random variables

Let(Xn)n≥1 be a sequence of independent identically distributed(i.i.d.) positive random variables with EX1 = μ,Var(X1) = σ2.In the present paper,we establish the moderate deviations principle for the products of partial sums(Πnk=1Sk/n!μn)1/(γbn√(2n))1where γ = σ/μ denotes the coefficient of variation and(bn) is the moderate deviations scale.     

An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions

We obtain an integro-local limit theorem for the sum S(n) = ξ(1)+?+ξ(n) of independent identically distributed random variables with distribution whose right tail varies regularly; i.e., it has the form P(ξ≥t) = t ^?β L(t) with β > 2 and some slowly varying function L(t). The theorem describes the asymptotic behavior on the whole positive half-axis of the probabilities P(S(n) ∈ [x, x + Δ)) as x → ∞ for a fixed Δ > 0; i.e., in the domain where the normal approximation applies, in the domain where S(n) is approximated by the distribution of its maximum term, as well as at the “junction” of these two domains.     

Large deviations for product of sums of random variables

In this paper, we obtain sample path and scalar large deviation principles for the product of sums of positive random variables. We study the case when the positive random variables are independent and identically distributed and bounded away from zero or the left tail decays to zero sufficiently fast. The explicit formula for the rate function of a scalar large deviation principle is given in the case when random variables are exponentially distributed.     

Integro-local theorems for sums of independent random vectors in the series scheme

Let S(n) = ξ(1)+?+ξ(n) be a sum of independent random vectors ξ(i) = ξ _(n)(i) with general distribution depending on a parameter n. We find sufficient conditions for the uniform version of the integro-local Stone theorem to hold for the asymptotics of the probability P(S(n) ∈ Δ[x), where Δ[x) is a cube with edge Δ and vertex at a point x.     

Large deviations of the first passage time for a random walk with semiexponentially distributed jumps

Suppose that ξ, ξ(1), ξ(2), ... are independent identically distributed random variables such that ?ξ is semiexponential; i.e., $P( - \xi \geqslant t) = e^{ - t^\beta L(t)} $ is a slowly varying function as t → ∞ possessing some smoothness properties. Let E ξ = 0, D ξ = 1, and S(k) = ξ(1) + ? + ξ(k). Given d > 0, define the first upcrossing time η ₊(u) = inf{k ≥ 1: S(k) + kd > u} at nonnegative level u ≥ 0 of the walk S(k) + kd with positive drift d > 0. We prove that, under general conditions, the following relation is valid for $u = (n) \in \left[ {0, dn - N_n \sqrt n } \right]$ : 0.1 $P(\eta + (u) > n) \sim \frac{{E\eta + (u)}}{n}P(S(n) \leqslant x) as n \to \infty $ , where x = u ? nd < 0 and an arbitrary fixed sequence N _n not exceeding $d\sqrt n $ tends to ∞. The conditions under which we prove (0.1) coincide exactly with the conditions under which the asymptotic behavior of the probability P(S(n) ≤ x) for $x \leqslant - \sqrt n $ was found in [1] (for $x \in \left[ { - \sqrt n ,0} \right]$ it follows from the central limit theorem).     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

Large deviations for sums of independent random variables with dominatedly varying tails

In this paper the large deviation results for partial and random sums Sn-ESn=n∑i=1Xi-n∑i=1EXi,n≥1;S(t)-ES(t)=N(t)∑i=1Xi-E(N(t)∑i=1Xi),t≥0are proved, where {N(t); t≥ 0} is a counting process of non-negative integer-valued random variables, and {Xn; n ≥ 1} are a sequence of independent non-negative random variables independent of {N(t); t ≥ 0}. These results extend and improve some known conclusions.     

Moderate deviations for stationary sequences of Hilbert-valued bounded random variables

In this paper, we derive the Moderate Deviation Principle for stationary sequences of bounded random variables with values in a Hilbert space. The conditions obtained are expressed in terms of martingale-type conditions. The main tools are martingale approximations and a new Hoeffding inequality for non-adapted sequences of Hilbert-valued random variables. Applications to Cramér-Von Mises statistics, functions of linear processes and stable Markov chains are given.     

Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

Let {_i} _i=1 be a sequence of independent identically distributed nonnegative random variables, S _n = ξ₁ + ? +ξ_n. Let Δ = (0, T] and x + Δ = (x, x + T]. We study the ratios of the probabilities P(S _n ε x + Δ)/P(ξ ₁ ε x + Δ) for all n and x. The estimates uniform in x for these ratios are known for the so-called Δ-subexponential distributions. Here we improve these estimates for two subclasses of Δ-subexponential distributions; one of them is a generalization of the well-known class LC to the case of the interval (0, T] with an arbitrary T ≤ ∞. Also, a characterization of the class LC is given.     

Large deviations for hitting times of a random walk in random environment on a strip

We consider a random walk in random environment on a strip, which is transient to the right. The random environment is stationary and ergodic. By the constructed enlarged random environment which was first introduced by Goldsheid （2008）, we obtain the large deviations conditioned on the environment （in the quenched case） for the hitting times of the random walk.     

长尾宽上限相依的不同分布的随机变量和的精确大偏差

研究了服从长尾分布族上的随机变量和的精确大偏差问题,其中假设代表索赔额的随机变量序列是一列宽上限相依的、不同分布的随机变量序列。在给定一些假设条件下,得到了部分和与随机和的两种一致渐近结论。     

Large deviations for sums of i.i.d. random compact sets

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets.

     

Extremes of Shepp statistics for Gaussian random walk

Let (ξ _i , i ≥ 1) be a sequence of independent standard normal random variables and let be the corresponding random walk. We study the renormalized Shepp statistic and determine asymptotic expressions for when u,N and T→ ∞ in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of when T,N→ ∞ and present corresponding normalization sequences.      

Small deviations of series of independent positive random variables with weights close to exponential

Let ξ, ξ₀, ξ₁, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series S = Σ _j=0 ^∞ a(j)ξ _j was studied under different assumptions on the rate of decrease of the probability ?(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ?(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ?(ξ < x) ～ bx ^α as x → 0, b > 0, a > 0, the asymptotics ?(S < x) is obtained in an explicit form up to the factor x ^o(1). Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].     

On sums of independent random variables without power moments

In 1952 Darling proved the limit theorem for the sums of independent identically distributed random variables without power moments under the functional normalization. This paper contains an alternative proof of Darling’s theorem, using the Laplace transform. Moreover, the asymptotic behavior of probabilities of large deviations is studied in the pattern under consideration.