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Register a new userProbabilistic proofs of large deviation results for sums of semiexponential random variables and explicit rate function at the transition
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Asymptotics deviation probabilities of the sum S n = X 1 + $times$ $times$ $times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated, in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when x n > n 1/2 (see, [13, 14]). In this paper, we derive rough asymptotics results (at logarithmic scale) with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition.
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Asymptotics deviation probabilities of the sum S n = X 1 + $times$ $times$ $times$ + X n of independent and identically distributed real-valued random variables have been extensively investigated , in particular when X 1 is not exponentially integrable. For instance, A.V. Nagaev formulated exact asymptotics results for P(S n > x n) when X 1 has a semiexponential distribution (see, [16, 17]). In the same setting, the authors of [4] derived deviation results at logarithmic scale with shorter proofs relying on classical tools of large deviation theory and expliciting the rate function at the transition. In this paper, we exhibit the same asymptotic behaviour for triangular arrays of semiexponentially distributed random variables, no more supposed absolutely continuous.
We derive properties of the rate function in Varadhans (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.
Let $(a_k)_{kinmathbb N}$ be a sequence of integers satisfying the Hadamard gap condition $a_{k+1}/a_k>q>1$ for all $kinmathbb N$, and let $$ S_n(omega) = sum_{k=1}^ncos(2pi a_k omega),qquad ninmathbb N,;omegain [0,1]. $$ The lacunary trigonometric sum $S_n$ is known to exhibit several properties typical for sums of independent random variables. In this paper we initiate the investigation of large deviation principles (LDPs) for $S_n$. Under the large gap condition $a_{k+1}/a_ktoinfty$, we prove that $(S_n/n)_{ninmathbb N}$ satisfies an LDP with speed $n$ and the same rate function $tilde{I}$ as for sums of independent random variables with the arcsine distribution, but show that the LDP may fail to hold when we only assume the Hadamard gap condition. However, we prove that in the special case $a_k=q^k$ for some $qin {2,3,ldots}$, $(S_n/n)_{ninmathbb N}$ satisfies an LDP with speed $n$ and a rate function $I_q$ different from $tilde{I}$. We also show that $I_q$ converges pointwise to $tilde I$ as $qtoinfty$ and construct a random perturbation $(a_k)_{kinmathbb N}$ of the sequence $(2^k)_{kinmathbb N}$ for which $a_{k+1}/a_kto 2$ as $ktoinfty$, but for which $(S_n/n)_{ninmathbb N}$ satisfies an LDP with the rate function $tilde{I}$ as in the independent case and not, as one might na{i}vely expect, with rate function $I_2$. We relate this fact to the number of solutions of certain Diophantine equations. Our results show that LDPs for lacunary trigonometric sums are sensitive to the arithmetic properties of $(a_k)_{kinmathbb N}$. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem by Salem and Zygmund or in the law of the iterated logarithm by Erdos and Gal. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems.
In this paper, we obtain some results on precise large deviations for non-random and random sums of widely dependent random variables with common dominatedly varying tail distribution or consistently varying tail distribution on $(-infty,infty)$. Then we apply the results to reinsurance and insurance and give some asymptotic estimates on proportional reinsurance, random-time ruin probability and the finite-time ruin probability.
We consider the problem of bounding large deviations for non-i.i.d. random variables that are allowed to have arbitrary dependencies. Previous works typically assumed a specific dependence structure, namely the existence of independent components. Bounds that depend on the degree of dependence between the observations have only been studied in the theory of mixing processes, where variables are time-ordered. Here, we introduce a new way of measuring dependences within an unordered set of variables. We prove concentration inequalities, that apply to any set of random variables, but benefit from the presence of weak dependencies. We also discuss applications and extensions of our results to related problems of machine learning and large deviations.
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