asymptotic behaviour of a sum

2012-06-02T17:08:46Z

I'd like to know the asymptotic behaviour as $N\to\infty$ of the following sum

$$ Z_N(x) := 2^{-N/2} \sum_{k=0}^{N/2} \frac{N!}{k! (N-2k)!} (N-1)^{-k} (\sqrt{2} x)^{N-2k} $$

in order to compute $p(x):=\lim_{N\to\infty} \frac{log(Z_N(x))}{N}$ with $x\geq0$ (I already know this limit exists) .

I found the lower bound $p(x)\geq\log(x)$, that by graphical simulations seems to be very good when $x$ is big enough.

Can you help me to compute $p(x)$?

Answer by Alex Becker for asymptotic behaviour of a sum

2012-06-02T19:31:50Z

Let $T_N(x)$ denote the largest term of the sum. Note that $$\frac{\log T_N(x)}{N}\leq \frac{\log (Z_N(x))}{N}\leq \frac{\log ((N/2)T_N(x))}{N}=\frac{\log T_N(x)}{N}+\frac{\log(N/2)}{N}$$ and that the limits of both sides are identical, so by Squeeze theorem $p(x)=\lim\limits_{N\to\infty} \frac{\log(T_N(x))}{N}$. To find the largest term, we4 want to minimize the expression $k!(N-2k)!(N-1)^k(\sqrt{2}x)^{2k-N}$. Note that $$\frac{(k+1)!(N-2k-2)!(N-1)^{k+1}(\sqrt{2}x)^{2k+2-N}}{k!(N-2k)!(N-1)^k(\sqrt{2}x)^{2k-N}}=\frac{2x^2(k+1)(N-1)}{(N-2k-1)(N-2k)}$$ and so we continue to make the denominator smaller by increasing $k$ so long as $2x^2(k+1)(N-1)<(N-2k-1)(N-2k)$. Solving for $k$ (and making approximations valid for large $N$) gives $$\begin{align}k &< \frac{N x^2+2N-x^2-1-\sqrt{N^2x^4+4N^2x^2-2Nx^4+2Nx^2+x^4-6x^2+1}}{4}\\ &< \frac{N x^2+ 2N-\sqrt{x^4+4x^2}N}{4}=\frac{2+x^2-\sqrt{x^4+4x^2}}{4}N\\ \end{align}$$ and so for large $N$ the largest term will be $k=f(x)N$, with $f(x)=\frac{2+x^2-\sqrt{x^4+4x^2}}{4}$. Note that we can assume equality (rather than accuracy to the nearest integer) without hurting our calculations since the limit exists and for any $\epsilon$ we have arbitrarily large $N$ such that this is within $\epsilon$ of an integer. Thus $$T_N(x) = 2^{-N/2}\frac{N!}{(f(x)N)!((1-2f(x))N!)}(N-1)^{-f(x)N}(\sqrt{2}x)^{(1-2f(x))N}$$ and so taking logarithms and applying Stirling's approximation ($\log(y!)\approx y\log(y)-y$) we get $$\begin{align}\log(T_N(x)) &= -\frac{N}{2}\log(2)+\log(N!)-\log((f(x)N)!)-\log((1-2f(x))N!)\\ & \;\;\;\;-f(x)N\log(N-1)+(1-2f(x))N\log(\sqrt{2}x)\\ &\approx -\frac{N}{2}\log(2)+N\log(N)-N-f(x)N\log(f(x)N)+f(x)N\\ & \;\;\;\;-(1-2f(x))N\log((1-2f(x))N)+(1-2f(x))N-f(x)N\log(N-1)\\ & \;\;\;\;+(1-2f(x))N\log(\sqrt{2}x)\\ \end{align}$$ and so (note we make the approximation $\log(N-1)\approx \log(N)$) $$\begin{align} \frac{\log(T_N(x))}{N}&\approx -\frac{\log(2)}{2}-f(x)+\log(N)-f(x)\log(f(x)N) \\ & \;\;\;\;-(1-2f(x))\log((1-2f(x))N)-f(x)\log(N-1)\\ & \;\;\;\;+(1-2f(x))\log(\sqrt{2}x)\\ &\approx -\frac{\log(2)}{2}-f(x)+\log(N)-f(x)\log(f(x))-f(x)\log(N) \\ & \;\;\;\;-(1-2f(x))\log(1-2f(x))-(1-2f(x))\log(N)-f(x)\log(N-1)\\ & \;\;\;\;+(1-2f(x))\log(\sqrt{2}x)\\ &\approx -\frac{\log(2)}{2}-f(x)-f(x)\log(f(x))-(1-2f(x))\log(1-2f(x))\\ & \;\;\;\;+(1-2f(x))\log(\sqrt{2}x)\\ \end{align}$$ thus $$\begin{align} p(x) &= \lim\limits_{N\to\infty} \frac{\log(T_N(x))}{N}\\ &\approx -\frac{\log(2)}{2}-f(x)(1+\log(f(x)))+(1-2f(x))(\log(\sqrt{2}x)-\log(1-2f(x)))\\ \end{align}$$ which should be quite close to the actual value due to the accuracy of Stirling's for large values.

asymptotic behaviour of a sum - MathOverflow

asymptotic behaviour of a sum

Answer by Alex Becker for asymptotic behaviour of a sum