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Let $\varepsilon_i$ be independent random variables such that $\mathbb{P}(\varepsilon_i = \pm 1)= 1/2$ and denote $W_n = \sum_{i=1}^{n}\varepsilon_i$. That is, $W_n$ is the simple random walk on $\mathbb{Z}$ started at $0$. Denote by $\tau_k$ the hitting time of $k\in \mathbb{N}$ (the smallest time $t$ when $W_t=x$). I am interested in the simple random walk that is stopped if it hits $k$. So let us define $$S_n = \sum_{i=1}^{\min(\tau_k, n)}\varepsilon_{i}.$$

Question. Does the generating function $f_h(t)=\mathbb{E}e^{hS_n}$ have a simple analytical expression and maybe it is known?

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Take any positive integers $n$ and $k$. Then
\begin{equation*} Ee^{hS_n}=E\exp\{hW_{\min(n,\tau_k)}\}=s_1+s_2, \tag{1}\label{1} \end{equation*} where \begin{equation*} s_1:=Ee^{hS_n}1(\tau_k\le n)=e^{hk}P(\tau_k\le n), \tag{2}\label{2} \end{equation*} \begin{equation*} s_2:=Ee^{hW_n}1(\tau_k>n) =\sum_{j=-n}^{k-1}e^{hj}P(\tau_k>n,W_n=j). \tag{3}\label{3} \end{equation*}

By the reflection principle, for each integer $j\le k$ \begin{equation*} P(\tau_k\le n,W_n=j)=P(W_n=2k-j) \tag{4}\label{4} \end{equation*} (see details on this below), so that \begin{equation*} P(\tau_k>n,W_n=j)=P(W_n=j)-P(W_n=2k-j) \tag{5}\label{5} \end{equation*} and \begin{equation*} \begin{aligned} P(\tau_k\le n)&=P(\tau_k\le n,W_n\le k)+P(\tau_k\le n,W_n>k) \\ &=\sum_{j\le k}P(\tau_k\le n,W_n=j)+P(W_n>k) \\ &=\sum_{j\le k}P(W_n=2k-j)+P(W_n>k) \\ &=P(W_n\ge k)+P(W_n>k). \end{aligned} \tag{6}\label{6} \end{equation*}

By \eqref{1}, \eqref{2}, the definition of $s_2$ in \eqref{3}, \eqref{6}, the latter equality in \eqref{3}, and \eqref{5}, \begin{equation*} \begin{aligned} Ee^{hS_n}&=e^{hk}P(\tau_k\le n)+Ee^{hW_n}1(\tau_k>n) \\ &=e^{hk}(P(W_n\ge k)+P(W_n>k)) \\ &+\sum_{j=-n}^{k-1}e^{hj}(P(W_n=j)-P(W_n=2k-j)). \end{aligned} \tag{7}\label{7} \end{equation*}

The latter expression can be re-expressed in terms of the c.d.f.'s of the binomial distributions with parameters $n,\dfrac12$ and with parameters $n,\dfrac{e^{2h}}{e^{2h}+1}$ (see details below), and hence in terms of the incomplete beta function (see e.g. this comment).


Details on \eqref{4}: For each integer $j\le k$ \begin{equation*} \begin{aligned} &P(\tau_k\le n,W_n=j) \\ &=\sum_{t=1}^n P(\tau_k=t,W_n=j) \\ &=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n=j) \\ &=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n-W_t=j-k) \\ &=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k)P(W_n-W_t=j-k) \\ &=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k)P(W_n-W_t=k-j) \\ &=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n-W_t=k-j) \\ &=\sum_{t=1}^n P(W_1<k,\dots,W_{t-1}<k,W_t=k,W_n=2k-j) \\ &=\sum_{t=1}^n P(\tau_k=t,W_n=2k-j) \\ &=P(\tau_k\le n,W_n=2k-j) \\ &=P(W_n=2k-j), \end{aligned} \end{equation*} since $2k-j\ge k$. $\quad\Box$

Details on re-expressing the latter expression in \eqref{7} in terms of the c.d.f.'s of the binomial distributions with parameters $n,\dfrac12$ and with parameters $n,\dfrac{e^{2h}}{e^{2h}+1}$: Note that $W_n=2B_n-n$, where $B_n$ a random variable with the binomial distribution with parameters $n,1/2$. Let $F_{n,p}$ denote the c.d.f. of the binomial distribution with parameters $n,p$. Then \begin{equation*} P(W_n\ge k)=P(W_n>k-1)=P(B_n>(n+k-1)/2)=1-F_{n,1/2}((n+k-1)/2) \end{equation*} and hence/similarly \begin{equation*} P(W_n>k)=1-F_{n,1/2}((n+k)/2). \end{equation*} Also, for $p_h:=\dfrac{e^{2h}}{e^{2h}+1}$ and any real $a$ and $b$ such that $a\le b$, \begin{equation*} \begin{aligned} &\sum_{a\le j<b}e^{hj}P(W_n=j) \\ &=\sum_{(n-b)/2<i\le(n-a)/2}e^{h(2i-n)}P(B_n=i) \\ &=\sum_{(n-b)/2<i\le(n-a)/2}e^{h(2i-n)}\binom ni 2^{-n} \\ &=\cosh^n h\,\sum_{(n-b)/2<i\le(n-a)/2}\binom ni p_h^i(1-p_h)^{n-i} \\ &=\cosh^n h\,\big(F_{n,p_h}((n-a)/2)-F_{n,p_h}((n-b)/2)\big). \end{aligned} \end{equation*} So, \begin{equation*} \begin{aligned} \sum_{j=-n}^{k-1}e^{hj}P(W_n=j) =\cosh^n h\,\big(1-F_{n,p_h}((n-k)/2)\big) \end{aligned} \end{equation*} and \begin{equation*} \begin{aligned} &\sum_{j=-n}^{k-1}e^{hj}P(W_n=2k-j) \\ &=\sum_{i=k+1}^{2k+n}e^{h(2k-i)}P(W_n=i) \\ &=\sum_{i=k+1}^n e^{h(2k-i)}P(W_n=i) \\ &=\sum_{i=k+1}^n e^{h(2k-i)}P(W_n=-i) \\ &=e^{2kh}\sum_{j=-n}^{-k-1} e^{hj}P(W_n=j) \\ &=e^{2kh}\cosh^n h\,\big(1-F_{n,p_h}((n+k)/2)\big). \end{aligned} \end{equation*} So, in view of \eqref{7}, indeed we can express $Ee^{hS_n}$ in terms of the c.d.f.'s of the binomial distributions with parameters $n,\dfrac12$ and with parameters $n,p_h=\dfrac{e^{2h}}{e^{2h}+1}$. $\quad\Box$

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  • $\begingroup$ Thank you a lot for your detailed answer! $\endgroup$
    – Ddzin
    Oct 29, 2023 at 19:26

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