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We consider the problem of constructing guarantee procedures of statistical inference with fixed minimal observation number n* for discrimination of two hypothesesH0 : θ ∈ [θ1, θ2] and H1 : θ /∈ [θ1, θ2]with a one-dimensional parameter under the so-called d-posterior approach. Here, constraints are placed on the conditional probabilities for the validity of one or another hypothesis under the condition that this hypothesis is rejected. We give an asymptotic formula for n* in a scheme with severe (tending to zero) constraints on these conditional probabilities of hypotheses. Earlier, Volodin and Novikov found a similar formula for discrimination of one-sided hypotheses. In the present paper, the proof of the asymptotic formula is carried out under weaker constraints on the probability model. The accuracy of our formula is illustrated numerically for some probability models. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic representation of the ruin probability for a random walk with negative drift when the upper bound of the strip tends to infinity. The result is expressed via distributions of the trajectory supremum and overshoot below negative level. In the subsequent corollaries, we use some known asymptotic formulas for the distribution of the supremum which are valid under various restrictions on the rate of decreasing at infinity for the distribution of jumps of a random walk. [ABSTRACT FROM AUTHOR]

We obtain an asymptotic representation of the ruin probability for a random walk with negative drift when the upper bound of the strip tends to infinity. The result is expressed via distributions of the trajectory supremum and overshoot below negative level. In the subsequent corollaries, we use some known asymptotic formulas for the distribution of the supremum which are valid under various restrictions on the rate of decreasing at infinity for the distribution of jumps of a random walk. [ABSTRACT FROM AUTHOR]

Let X,Xi, i ≧ 1, be a sequence of independent and identically distributed random vectors in Rd. Consider the partial sum Sn ≔ X1 + ⋯ + Xn. Under some regularity conditions on the distribution of X, we obtain an asymptotic formula for P{Sn ϵ nA}, where A is an arbitrary Borel set. Several corollaries follow, one of which asserts that, under the same regularity conditions, for any Borel set A, limn→∞ n-1 log P{Sn ϵ nA} = -I(A), where I is a large deviation functional. We also prove a multidimensional Laplace-type approximation that allows an explicit calculation of the sharp large deviation probability typically when the set A has a smooth boundary. [ABSTRACT FROM AUTHOR]