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Expansions for random walks conditioned to stay positive

Denis Denisov, Alexander Tarasov, Vitali Wachtel

Abstract

We consider a one-dimensional random walk $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf P(τ_x>n)$ of the first passage times $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ for $\ x\ge0.$ We also derive asymptotic expansion for local probabilities $\mathbf P(S_n=x,τ_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.

Expansions for random walks conditioned to stay positive

Abstract

We consider a one-dimensional random walk with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution of the first passage times for We also derive asymptotic expansion for local probabilities . Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.
Paper Structure (19 sections, 29 theorems, 420 equations)

This paper contains 19 sections, 29 theorems, 420 equations.

Table of Contents

  1. Introduction.
  2. Some properties of binomial coefficients
  3. Asymptotic expansions in the central limit theorem
  4. Asymptotic expansions for $\mathbf P(S_n\le x)$
  5. Special algebra of generating functions
  6. Asymptotic expansions for $\tau_0$
  7. Proof of Theorem \ref{['thm:tau_0']}
  8. Calculation of the first two correction coefficients
  9. Expansions for conditioned probabilities
  10. Proof of Theorem \ref{['thm:cond_prob']} in the case of fixed $x$
  11. Proof of Theorem \ref{['thm:cond_prob']} for growing $x$.
  12. Asymptotic expansions for $\mathbf P(S_n=x,\overline{\tau}_0>n)$.
  13. Asymptotic expansions for $\tau_x$.
  14. Left-continuous case.
  15. General case
  16. ...and 4 more sections

Key Result

Theorem 1

Assume that $\mathbf E|X_1|^{r+3}$ is finite for some integer $r\ge0$. Assume also that either the distribution of $X_1$ is lattice with maximal span $1$ or it holds $\lim\sup_{|t|\to\infty}\left|\mathbf E~e^{itX_1}\right|<1$. Then there exist numbers $\nu_0,\nu_1,\ldots,\nu_{\lfloor r/2\rfloor}$ su where $\delta_r = \mathds{1}_{\{r \text{ is odd} \}}$. Furthermore, there exist numbers $\overline{

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 7
  • proof
  • Proposition 8
  • Proposition 9
  • ...and 45 more