# Conditioned limit theorems for random walks with negative drift

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## Summary

In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if *S*_{n} is a random walk with negative mean and finite variance then there is a constant *α* so that (*S*_{[n.]}/α*n*^{1/2}¦*N*>*n*) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if *ES*_{1}=−a<0, *ES* _{1} ^{2} <∞, and there is a slowly varying function *L* so that *P(S*_{1}>*x*)∼*x*^{−q} L(x) as *x→∞* then (*S*_{[n.]}/*n*¦*S*_{ n }>0) and (*S*_{[n.]}/*n*¦*N*>*n*) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1−(*x/a*)^{−q})^{+}) and are otherwise linear with slope −*a*. The jump occurs at a uniformly distributed time in the first case and at *t*=0 in the second.

## Keywords

Stochastic Process Random Walk Probability Theory Limit Theorem Mathematical Biology## Preview

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## References

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